If we're speaking of the western European tradition: I have an older answer that gives one answer to this question in response to:
I have added a section about women's same-sex activity!
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None. At all.
Scholars have long recognized the twelfth century as a pivotal moment in the intellectual history of sodomy in the west (and of pretty much everything, to be fair). Although there are scattered references to "sex as the sodomites do"--which is to say, anal intercourse regardless of the sex/gender of people involved--as definitely a sin and a very bad thing in earlier medieval sources, it's really in the 12th century that theologians and the new specialty of canon lawyers make it a particular point of focus and bitter opprobrium.
However, into the early 13th century, the primary target of this zeal was clergy. Accusations of heresy and sodomy went hand-in-hand in the Languedoc in the run-up to the Albigensian Crusade (against the people the Church deemed Cathars). Lateran III (Church council) in 1179 targeted clergy who violate clerical celibacy with either marriage or the "sin against nature": sodomy, since anal intercourse can never lead to reproduction.
Church councils in France in 1212 and 1214 also impressed the evils of sodomy among the clergy. The only time laymen were mentioned among these was in 1179, and even then, the recommendation was excommunication.
But the 13th century was its own turning point in the intertwined history of religion and sexuality. The Church had quite successfully used control over marriage practices (such as through laws on consanguinity) as a spearhead in entrenching itself as a power player in profane society. At Lateran IV in 1215 and with the introduction of new orders of preachers (especially the Franciscans and Dominicans), the Church committed itself to shepherding workaday laity more closely through life and to salvation, including through religious instruction and moral discipline. The diffusion of actual practice out of the ideal, including the most important points of "every Christian of both sexes" must confess their sins once a year to their parish priest and then receive the Eucharist at Easter, took time to filter downwards.
But pastoral theologians were enormously concerned with regulating sexual practices. Handbooks for confessors, arranged according to the seven deadly vices, devote vastly more space to sins under the umbrella of luxuria than the other six (although ira, wrath, gets a fair amount of play).
And intriguingly, some of the summa confessorum authors had a particular emphasis on sodomy. Far from the times of Aelred of Rievaulx not even daring to mention the sin directly for fear of accidentally corrupting the mind of someone who'd never thought of non-PIV sex, Paul of Hungary's summa spends roughly 100 times as much space on sodomy as on other sins of lust.
But it's noteworthy that Paul is still an extreme example at this stage. And just like scholars assume annual participation in confession and the Eucharist was hardly universal immediately in 1215, actual attempts to control lay sexuality beyond basic issues involving marriage in both canon and civic courts were very slow in coming.
Italy led the way. In 1250, Bologna decreed that the punishment for sodomy was exile, but the banished could petition for permission to return. In 1259, the city rescinded the possibility of forgiveness. Only in 1288 was sodomy declared a crime deserving execution. Contemporary German law codes were still ignoring sodomy as a crime.
In France, meanwhile, sodomy was apparently a possible crime by 1270. However, there were no prosecutions, convictions, or executions until the reign of Philip V (1316-1322; note that I am not counting the Templars here, since the sodomy accusations there were sort of an "...and the kitchen sink" kind of deal), during which there was a whopping...one.
One addition I do want to make this time. It's kind of a truism that sodomy, homosexuality, homophobia and so forth were in practice limited to men. This is not entirely true. In the Middle Ages, there are some in-retrospect-hilarious examples of texts like the Ancrenne Wisse, an early 13th century guide to religious life for women.
As Jane Bliss (real name) points out, the text contains a number of oblique references to both masturbation and lesbian sex ("it was with...an innocent creature" "a woman such as myself") as sins. But, as with Aelred, the author cannot/does not want to say anything explicitly for fear of putting ideas into women's heads.
I'm sure we all know exactly what to think about that.
But while the Ancrenne Wisse became a devotional text for women to study, the broader social and legal consequences for same-sex sex also turned against women. Helmut Puff has published a few times on the case of Katharina Hetzendorfer in 1477 Speyer. She was convicted and executed, in the manner typically used on women, for engaging in sex with women "like a man," using a dildo.
It's nobody's place to speculate on gender identities or even what that would mean in the Middle Ages (a topic of much scholarly discussion right now). But it shows first of all the roots of this strand of homophobia, at least, in gender-lines transgression. It also shows that Hetzendorfer here was seen as a woman having sex with other women.
But as an additional layer of interest: there was apparently no name for this crime, under the law. But she was still drowned for it.
England, meanwhile? England appears to have been the least concerned with sodomy as a whole during the High Middle Ages. Not only was sodomy not on the law books in the 13th century, but even during the 15th and 16th centuries when convicted sodomites (mostly men, but a handful of women as well) were burned to death with some regularity in many countries (especially Italy)--executions in England for the sin against nature were surprisingly rare.
We have no surviving peasant voices from the Middle Ages, so we cannot reconstruct the fama (or infamy, in this case) associated with sodomy in 13th century English villages. This makes it difficult to talk about potential social consequences on the daily life level. However, from an official standpoint, execution at this time was not an option.
This will make the rumors about Edward II all the more interesting and the evidence all the more ambiguous. The existence of slander against someone through accusations of same-sex sex suggests less about the people involved, and much more about the culture that considers it a damning insult.
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P.S. Please excuse my moderation comments elsewhere in this thread. Normally the AH mod team has a policy against moderating in threads where we are actively participating, but I am/was the only mod around at the time.
So the fabulous /u/sunagainstgold has us covered for the Middle Ages, but your question touches on classical Greece and Rome directly, so its worth mentioning what happened in these places between the height of the Roman Empire and the Middle Ages. In between this space lies "Late Antiquity", and its when things just kinds stopped making sense.
The defining change of the Roman Empire from Classical to Late Antiquity was the conversion of the government, if not the people at large just yet, to Christianity. Along with this change Kyle Harper argues in From Shame to Sin that attitudes towards homosexual behavior between men were fundamentally altered.
Firstly, Harper argues that we need to understand same sex relationships between men in the high Empire are not necessarily the same as a free acceptance of homosexuality. Instead the "acceptable" forms of same sex expression in the Late Roman world were invariably between masters and slaves, not free born men (and I do mean men, women feature between very rarely and not at all in ancient discourses on same sex relations). Indeed, seducing a free born man into sex with a man was a severe crime in Roman Law even before conversion to Christianity, this was not the Greek approach between men that was seen in Athens and the like in the 4th century BC. Secondly the survival of these master slave relations in the Roman Empire was also predicated on the widespread sexual availability of slaves through brothels. This point is worth underscoring. Without the widespread sexual availability of slaves to their masters, through both private ownership as well as through the sex trade, there was no Roman approach to homosexuality.
When Christianity became the prevailing ideology of the empire in the 4th century sexual slavery and public prostitution were heavily frowned upon, as in "could result in public execution" levels of frowned on. Over the course of the next several centuries various emperors would outlaw or suppress both sexual slavery and same sex relationships. The emperor Theodosius oversaw the public burning of male prostitutes in Rome and later Justinian would pass harsh laws condemning the sale of people into sexual slavery against their will. He condemned people participating in prostitution as well but the prevailing idea was that if you weren't being forced into the life you were legally culpable, so non-enslaved prostitutes were also targeted by Justinian's reforms.
These legal developments went hand in hand with Christian developments of the will and the relationship between sexuality and the will. Harper concludes that over time the Christian ideas and messages seeped into the broader society through a top down implementation of strict anti-same sex relationships that ultimately derived from the letters of Paul that were used by rabble rousing bishops in Constantinople for example.
Over time however this focus lost much of its impact and focus and as we enter into the Medieval period Roman law collapses in much of their former territory and it is more difficult to understand the approaches that these successor kingdoms (with the exception of the Eastern Romans) took towards homosexual behavior. In some non-post-Roman contexts there is still some evidence, such as the Norse people of the Early Middle Ages who also seemed to share a dismal approach towards same sex relationships. ( https://www.reddit.com/r/AskHistorians/comments/5vb166/how_did_pagan_vikings_in_the_813th_centuries_view/de1bo9q/ here is a post on this topic from the aforementioned /u/sunagainstgold )
Modern western homophobia is a distinctive modern phenomenon, that must be distinguished from older moral objections to same-sex relations. I'm not aware of any historical parallels for modern homophobia (although I don't claim expertise on all societies and periods!)
Modern homophobia is distinctive in that it pathologizes homosexuality: in other words, homophobes think that there's something wrong with homosexual people, and with the very experience of homosexual desire. We see this in the homophobic use of words like unnatural or perverted. At its most extreme, homosexuality was understood to be a form of mental illness: it was listed in the American Psychiatric Association's Diagnostic and Statistical Manual of Mental Disorders (DSM) until 1973. Indeed, this modern concept of homophobia developed in tandem with the emergence of psychiatry as a science and branch of medicine in the 19th century, along with Victorian morality. It then spread to much of the rest of the world via colonialism and popular culture.
By contrast, most forms of hostility or prejudice from periods before the 19th century saw the act of homosexual sex as morally wrong, but did not insist that people who experienced homosexual desire were suffering from a disorder. It was analogous to views towards adultery and extra-marital sex. People thought that a person who committed adultery had done something wrong, but they didn't think it was perverse that a person would be sexually attracted to someone other than their spouse. On the contrary, sexual desire for people other than one's spouse was considered totally normal, but acting on that desire was considered wrong. In fact the experience of desire is what makes monogamy or celibacy meritorious in these value systems: you control your emotions and passions rather than being led by them. Pre-modern attitudes towards homosexuality were broadly similar: if a person experiences homosexual desire, he or she should not act on it, but there is nothing inherently wrong with the desire itself.
Within that basic framework, different societies had a range of different attitudes to same-sex desire. I can describe the one I know best, which is the Islamic Middle East from the classical Islamic period (9th/10th century CE) up to the 19th century. I'll only talk about men's sexual desire, which we know far better than women's because most of the sources were written by men.
Today you'll find many conservative Muslims, and also western Islamophobes, who insist that sharia law bans homosexuality. But in fact what sharia law bans is sodomy. And like the analogies I used above, but unlike modern homophobia, it is focused on the act rather than the desire. So homosexual sodomy was a criminal offense, and extra-marital heterosexual sex was also a criminal offense. But in neither case did sharia law say that there was anything wrong with experiencing the desire. In fact, throughout much of this period men wrote openly and at length about their sexual attraction to men and women, and there is little indication of any embarrassment, or sense that attraction to men was deviant, or even that there was a hierarchy between heterosexual and homosexual desire. In poetry, some of the standard images used to describe the beauty of the beloved were mainly applicable to men. For example, poets often referred to the beloved's "cheek-down," i.e. of an adolescent boy who does not yet have a beard. In Turkic languages there is no grammatical gender - i.e. there is no distinction between he and she or his and her - and poets loved to play around with the ambiguity this provided. Plenty of elite, socially respectable men wrote about their homosexual desire (including statesmen, judges, scholars, businessmen, etc., since everyone was a poet in those days). In fact, although it's impossible to be sure because of the ambiguity of Turkish, some historians claim that in the Ottoman Empire the poetic paradigm for desire was homosexual - i.e. the basic model of desire and longing that poets worked with, with all its associated imagery and metaphors for beauty, was a man in love with an adolescent boy, and desire for a woman was a variation on the theme. As long as homosexual desire was unconsummated, nobody thought it the least bit problematic or strange. Plenty of these men also engaged in homosexual relations - in private homes and in bathhouses. This was officially a criminal offense, and there stern religious types who thought it a was very serious offense (although, as with any sexual offense that is committed in private, prosecutions were rare). Then there were other people who thought this was naughty in a kind of "politician caught with his pants down" way, and others who thought it was fine and people should mind their own business. But to reiterate, those who thought it wrong thought that the sexual acts were morally wrong, not that the desire was deviant. So this isn't equivalent to modern homophobia - in fact it's not really a "phobia" at all.
There's a wrinkle here because some kinds of homosexual relationship were considered deviant or unnatural. There was a distinction between penetrating and being penetrated, and this was bound up with status and sexism. The elite men writing about their homosexual desire were considered "normal" as long as their desire was to penetrate their beloved. If they desired to have their beloved penetrate them, that was another matter. This was because being penetrated was associated with submission, and with being in the position of a woman, so it was incompatible with ideas of masculinity. Now obviously there has to be another man on the other side of the encounter, and it might seem incoherent for one side to be thought normal and the other not. But in the paradigm of this kind of relationship, the penetrated beloved is much younger - hence the poets' focus on their lovers' cheek-down. Submissively receiving penetration was thought acceptable during a particular stage of a man's life: adolescence, before he became a full man (and full manliness was strongly associated with beards, which is why all Middle Eastern men wore beards until the mid-19th century - shaving it off was seen as ludicrous). But once one was a full man, it became shameful. So in this way some kinds of homosexual desire were pathologized, but it's a different kind of pathologizing to modern homophobia. It wasn't about the sex of the man's partner, but about the man not dominating his partner in the way proper men were expected to (and so being penetrated by a woman was also considered weird). I'm no expert but I believe that attitudes to homosexuality in ancient Greece were similarly inflected by a status distinction between penetrator and penetrated. I would argue that this is more misogynistic than homophobic.
And there's one last point which makes this even more complicated. As well as heterosexuality and homosexuality, our very concepts of love and sex are also historically contingent. Put simply, in today's society we bundle physical attraction, emotional bonds and support, sex, economic relationship and raising a family together into the model of monogamous, companionate marriage. Even as we become more tolerant of relationships other than heterosexual marriage - e.g. gay marriage, and cohabiting but unmarried couples whether mixed- or same-sex - these are still essentially modeled on marriage, just without the precise legal requirements we inherited from the church. In past societies marriage was mostly an economic institution. But apart from marriage, at the most basic level love and sex weren't always connected in the same way as our idea of romantic love. C.S. Lewis wrote an interesting book about medieval French poetry, called The Allegory of Love, in which he argued that our notion of romantic love only came into being in the high Middle Ages. In earlier times, a man might be sexually attracted to women, and have sex with women, but his strongest, closest and most dependent emotional relationships were with other men and were non-sexual. In early medieval Europe, the paradigm for love was the bond between a man and his fellow warriors; then around the 12th century writers began to apply similar kinds of affection and emotional connection to sexually desirable and subordinate women.
In conclusion, the key point is that homophobia didn't emerge until the 19th century, and neither did homosexuality in the way we understand it today. There were laws prohibiting homosexual conduct - mostly sodomy - in many past societies (possibly including Greece & Rome, I'm not sure), but that isn't the same as homophobia which pathologizes homosexual desire. It's also misleading to say that homosexuality was thought normal in ancient Greece and Rome. They didn't think homosexuality as we understand it was normal: they didn't allow gay marriage legally, and they didn't think it was normal for someone to form a long-term sexual relationship with someone of the same sex instead of getting married and raising a family. Where they differed from modern homophobia is in thinking it normal and acceptable for people to have some kinds of same-sex encounters as well as marrying (whether before marriage or at the same time). And in this way ancient Greece and Rome were similar to most pre-modern societies, which neither pathologized homosexual desire nor linked love, sex and marriage in the same way we do.