From what I understand of the Mongolian Empire in which you refer, there were indeed specific tenants put into place that would have made homosexuality a view that was not tolerated.

In the book, Genghis Khan: The Emperor of All Men, there is a basic list of laws in place to govern the Mongol people, and in fact one such law was "Whoever is guilty of sodomy is also to be put to death".

Despite religious tolerance, homosexuality was something that would not have went over well with the Mongols.

I'm not at home right now so I can't pick out the exact books I'm using as second-hand sources, but I will be sourcing multiple ancient texts. If the mods feel that doesn't suffice, please remove this response and I'll try and find the other sources later. I don't know much about Mongolian culture besides what I've picked up from a couple books on Genghis Khan, but I actually wanted to address something you said as an aside.

I know LGBT rights and acceptance has been a more modern development

The history of human sexuality is a very interesting topic, especially concerning homosexuality. In most cultures for most of history, "being gay" wasn't something you were, it was something you did, an action, not an orientation. The popularly quoted section of Leviticus bans sodomy(an action), not preferring your own gender(an orientation). The associated flamboyancy or stereotypical effeminacy is entirely separate. What's also very interesting to note is that almost every topic of homosexuality in ancient times is only concerned with male sexuality, so I actually can't comment much on female sexuality. Even in some Arab countries that put people to death for being gay, it applies much more harshly to men since it's actually about sodomy.

As a global movement, LGBT rights is a recent trend, but in various cultures and periods of history it's been tolerated or even accepted as normal private actions. If you've ever seen some of the Hindu temples in India like the Meenakshi temple, they are covered with depictions of homosexual relations. Sexual orientation wasn't really a concept, and it was understood that when men got together they would have sex with each other but it wasn't seen as being different. It's important to note that men having sex with each other was not associated with effeminacy/flamboyancy, which was seen as weird or different. Hinduism (which isn't an accurate term to use in pre-colonial times) also acknowledges non-binary genders such as Hijra or Kotis (referred to collectively as "third-gender"). There's multiple characters in the Mahabharata who change genders like Shikhandi who is born female but identifies as male gender and eventually marries a woman. Bahuchara is the god of transsexuality, worshipped by Hijras. Hell, even Shiva is specifically a very androgynous figure. Ancient Hinduism recognizes genders but isn't concerned with sexual orientations. Gods change genders all the time.

Another Indian religion, Buddhism, also never really cared about homosexuality. Buddha is often depicted as asexual. It is important to note, though, that ancient Buddhist texts identified promiscuous homosexuals (primarily prostitutes) as a separate class and has generally had a negative view of them. My understanding of it is that it has more to do with their lust/desire for base needs that brings that view, not the fact that they are gay, as is described in the Vinaya. While there are certainly many Buddhist writings that condemn homosexuality as a behavior, East Asian Buddhism is even more tolerant. Francis Cabral, a 16th century priest, traveled to Japan and wrote that he found them to be "abominations" because of their acceptance and openness with bisexuality and homosexuality. A medieval Buddhist monk (can't remember who) wrote about how monks would have sex with each other because they were not allowed to have relations with women or "third-genders", to the point where being celibate meant that you were not much more than a piece of ass (this is the literal translation, not my own addition). China has had similar views, though there are older writings that believe homosexuality leads to rebirth as a lower organism or something of that nature. On a funnier note, North Korea's official website has a page on LGBT. They acknowledge that it's perfectly natural and that they don't discriminate like the Western world, but marriages need to be heterosexual so that their population can grow. This is not an uncommon view (I might go as far as to say it's been the dominant view for most of history).

Aside from the eastern hemisphere, which has only recently had more conservative views of sexuality due to western influence, you also have to remember ancient Greece and Rome. Aristophanes writes about a practice where a man takes an adolescent boy under his wing as a student and lover, as part of the boy's education. It's very possible that this was only limited to the upper class, as it's mentioned as a form of education/pedagogy. While the movie 300 would lead you to believe differently, this was especially prevalent in Sparta. In Rome, taking the passive role was seen as demeaning, so a Roman citizen would have a male slave or prostitute as a lover. Taking a freeborn citizen as a lover was strictly off-limits as it was considered a violation of their rights. (I wanna say the source for this is Amy Richlin?).

But anyways, as you can see, LGBT acceptance is not a recent development. I'd even go as far as to say the condemnation of it is primarily an Abrahamic concept, at least as far as the documented world goes. I can't say much about sexuality in pre-Columbian America or sub-Saharan Africa since most of their written history is from Christian outsiders.