I just finished a thoroughly frustrating historical novel dealing with the reign of Richard I. After doing a quick bout of research to seperate fact from fiction I'm still thoroughly confused by the "was he gay?" debate. A lot of people seem to think the claim is silly and based on insufficient knowledge of the time ("zOMG it says sodomy/he shared a bed with a dude"). On the other hand a lot of people on Team Heterosexual base their claim on the fact that he did have intercourse with women (because bisexuality doesn't exist) and seem to disregard some of the sources cited by those who think he might have been gay/bi.
(Should add that's based on very shallow internet research, and that I'm as tired of "was X gay" as you probably are.)
This is one of those topics that is subject to endless speculation because queer studies has been a "hot" academic topic for the past few decades. The two main stories surrounding Richard I are during his travels to the Holy Land, he and Philip Augustus stopped in Italy to confess. The two kings tried to one up themselves and Richard one by saying "I bone dudes!". The other is that after he was captured on his return from the Holy Land, he got word to his supporters via his personal bard, who he supposedly sang to from his tower prison cell. This bard is considered by some to be Richard's lover.
It should be noted that while there is some debate, John Gillingham, arguable the leading expert on the Angevin period has discounted any claims to Richard's homosexuality in John Gillingham, "Richard I and Berengaria of Navarre," Bulletin of the Institute of Historical Research, 53 (1980): 157-73 and 169-71. Others, like William Barlow and John Boswell, who assumed Richard's homosexuality without concrete documentary proof, have been shot down by prominent scholars such as Elizabeth A.R. Brown in Ritual Brotherhood in Western Medieval Europe, Traditio, Vol. 52 (1997), pp. 357-381. There were laws against homosexuality, both sacred and secular, but it wouldn't be unusual for someone as powerful as Richard I to get around them.
It is also worth noting Cicero's work De Amicitia (On Friendship), which goes way beyond a bro-mance. To Cicero and the Romans, male companionship was preferable to female companionship because men were smarter and more perfectly designed. Women were good, but only for feminine things like baby-making, weaving, and polite conversation. Cicero's works were well-known in the middle ages and there many Roman holdovers regarding social niceties, so it would not be unreasonable to assume that medieval people held a similar view of male friendship - a good parallel would be the medieval (secular) curia or curia regis.
See these books for more info:
(1) Elizabeth Keiser. Courtly desire and medieval homophobia : the legitimation of sexual pleasure in Cleanness and its contexts (1997)
(2) Joan Cadden. Nothing natural is shameful : sodomy and science in late medieval Europe (2013)
(3) Dawn Hadley. Masculinity in medieval Europe (1998)
(4) James Neill. The origins and role of same-sex relations in human societies (2009)
Hope this helps a little. Happy Reading!