During the course of the first centuries of the first millenium, Roman attitudes toward same-sex relationships gradually grew more conservative. Through a series of laws under multiple Christian Roman emperors, homosexuality transformed from something rarely mentioned to a sin punishable by burning. Lets go through these changes one by one--

• Phillip the Arab (244-249): Phillip was supposedly a Christian convert (making him the first Christian emperor). According to John Chrysostom he is said to have attempted to join the Easter celebrations at Antioch. Chrysostom's account of his faith is corroborated by Eusebius ( Ecc. Hist. VI.34 ), but he was likely simply lenient to Christians rather than an active Christian. Nevertheless, we shall examine him as a "Christian" emperor. Phillip outlawed male prostitution, beginning the campaign against homosexuality (especially pertaining to male/male relationships).
• General Events of the Social Crisis: The social crisis of the 3rd century (marked by economic turmoil and a series of very short reigns by emperors), saw the outlaw of same sex marriages as well as several other changes relating to sexuality in the Empire.
• Changes under the Theodosian Code: The emperors of the 4th century, made the coupling of men with other men a crime punishable by sword under the Theodosian Code. Codex Theodosianus 9.7.3 reads as follows.

9.7.3
Impp. Constantius et Constans aa. ad populum.
Cum vir nubit in feminam, femina viros proiectura quid cupiat, ubi sexus perdidit locum, ubi scelus est id, quod non proficit scire, ubi venus mutatur in alteram formam, ubi amor quaeritur nec videtur, iubemus insurgere leges, armari iura gladio ultore, ut exquisitis poenis subdantur infames, qui sunt vel qui futuri sunt rei.
Translated as follows:
Where a man is married to the woman, a woman of the men, the total projection of what he wished, where the sex has lost its place, where there is the crime of this kind is that which does not advance, to know, to where the Venus of it is changed into the one form, where the love of the question is neither hath he seen, we order to rise up to the laws of, to be armed for the rights of the sword, the avenger, so that the infamous subjected to cruel penalties that are either those that are real.
to be noted: "woman" here is meant as an insult to a feminine man, not a woman in gender, but a "woman" in action

• Changes in the Sixth Century: Emperor Justinian (a notoriously Christian Emperor) took these steps against homosexual relationships further. Justinian seems to have taken the issue of homosexuality in a specifically Christian context. Procopius mentions these specific tactics in his Secret history.

Whether or not this change in policy was directly to do with the rise of Christianity is still fiercely contested by scholars. While a few emperors seem to have taken these issues more seriously (Justinian), there are numerous other factors involved. The strength of the Church at the time as well as disentangling the promotion of abstinence/asceticism vs. discouraging homosexuality explicitly are topics that are still debated. This debate is summarized below.

Early Christian writings were generally antagonistic to homosexuality. However, this antagonism was not restricted to Christian thought. Although John Chrysostom was among the most vehement of the early fathers of the church in his denunciations of homosexuality (Boswell 1980, pp. 359-60), his somewhat older contemporary Libanius, a pagan born in Antioch, expressed similar feelings with equal forcefulness (Jones 1961, p. 972). Boswell notes that in the West, the Christian church was the only organized entity to survive the German invasions. It thus became "the conduit through which the narrower morality of the later Empire reached Europe. It was not, however, the author of this morality. The dissolution of the urban society of Rome and the ascendance of less tolerant political and ethical leadership occasioned a steady restriction of sexual freedom which transcended credal boundaries. . . . All the organized philosophical traditions of the West grew increasingly intolerant of sexual pleasure under the later Empire, and it is often impossible to distinguish Christian ethical precepts from those of pagan philosophy during the period" (1980, pp. 127- 28)

Greenberg et. al. also argue that early Christianity wasn't solely against homosexual relationships rather sexual practice in general.

Moreover, Christian hostility to homosexuality was not directed against that alone, but toward all forms of sexual activity. Although the New Testament did not look favorably on sexual expression, the leaders of the early church gave sex much greater attention and rejected it far more passionately and completely. Virtually all the church fathers- Gregory of Naz- ianzus, Gregory of Nyssa, John Chrysostom, Ambrose, Augustine, Jerome- praised virginity and looked on sex with horror (de Mendieta 1955; Dodds 1965). Tertullian regarded unchastity as worse than death (Bul- lough 1973, p. 98). In the early Syrian church, only unmarried Christians could be baptized (V66bius 1951), and some Western bishops of the second century made continence compulsory for church members (Grant 1970, p. 271). A number of Christian writers, including Eustathius of Sebastia, a bishop of the mid-4th century, held that married people could not be saved. Augustine considered sexual pleasure within marriage to be sinful, even though intercourse was nonetheless redeemed by the desire for children and by the sacramental character of marriage (Erickson 1976). The 2ndcentury apocryphal Acts of the Apostles maintained that married persons should refrain from sex (Davies 1980, pp. 32-33), and "spiritual" (celibate) marriages were not uncommon among early Christians

In sum, while we are certain that there were changes explicitly made regarding homosexuality, it is still debated how significant of a role Christianity played in bringing these changes about.

Theodosian Code: here

Sources Consulted:

Greenberg, David F., and Marcia H. Bystryn. “Christian Intolerance of Homosexuality.” American Journal of Sociology, vol. 88, no. 3, 1982, pp. 515–548. JSTOR, www.jstor.org/stable/2779118. Accessed 10 July 2020.

Christianity, Social Tolerance, and Homosexuality: Gay People in Western Europe from the Beginning of the Christian Era to the Fourteenth Century(University of Chicago Press, 1980)

Additional Sources:

Christopher Records, "When Sex Has Lost its Significance: Homosexuality, Society, and Roman Law in the 4th Century", in UCR Undergraduate Research Journal, Volume IV (June 2010)

Michael Brinkschröde, "Christian Homophobia: Four Central Discourses," in Combatting Homophobia, p. 166.