Noted Objectivist scholar Chris Matthew Sciabarra wrote a famous monograph on this topic. I have not read it, but accounts and reviews indicate its focus is historical, not doctrinal. This essay will seek to show that Objectivism and Libertarianism, as philosophy, hold appeal precisely because of their ideas.
In Atlas Shrugged, John Galt is at one point described as "the man without pain or fear or guilt." This is every LGBT's dream and goal with respect to how we feel about our sexuality. Atlas Shrugged, in the theory of "white blackmail," says that the establishment takes our virtues and tricks us into regarding them as sins, then uses our ensuing guilt to manipulate and rule us. While Rand focused on the sin of greed, logically this is equally true for the sin of lust.
In The Fountainhead, Roark is so committed to being an architect that he will let no one and nothing stop him. The stigma and scorn of society, which criticized and disdained his buildings throughout much of the novel, does not bother him. He tells Toohey, the villain who embodies society's disapproval, "but I don't think of you," when asked how much he hates him. Rand at one point says Roark could walk down a street naked and really not notice if anyone cared.
This disregard for social disapproval is central to the gay agenda. Homophobia does exist, equally among conservatives, moderates, liberals, and, yes, libertarians. LGBTs face scorn and hatred and shame and stigma and the pressure to be closeted. The attitude of not giving a damn what other people think of you embodied in The Fountainhead is a useful tool for gay pride, not politically but rather psychologically.
Rand was herself interesting from an LGBT point of view. Her own sexual repression, and the repression of her 1960s followers, is well documented. She seems to have repressed her sexual desires her entire life and then misdirected them at Nathaniel Branden, the much-younger guy she had an affair with as an older woman. She is also known to have had special praise for Mises when he called her "one of the most courageous men in America." Barbara Branden in her biography of Ayn Rand said that Rand was happy that Mises had called her a man. We can wonder what that meant.
Libertarianism is also a good political philosophy for LGBTs, although many gays favor leftism.
"Live and let live," the sum of our philosophy, means every LGBT gets the freedom to do what they want, even if some members of society regard it as evil. In contrast, gay socialists want to force everyone to like gays by outlawing homophobia. This goal of everyone loving gays is impractical and unachievable. We libertarians would agree to outlaw homophobic violence, but we let each person have the freedom to believe what they want. Freedom treats people with respect, and you have the freedom to associate with those who accept you, and avoid the ones that don't. As an extension of this, I have advocated that each church, temple and synagogue be free to choose to recognize or reject gay marriage as they decide according to their own moral compass, with government marriage licenses abolished.
"Stay out of my bedroom" and "mind your own business," along with "live and let live," are key mottoes and mantras for both LGBTs and libertarians. These phrases show that gay political activism and libertarian ideas can share a point of view.
Here I have made a compelling argument that LGBT (and Queer, Non-Binary, Pansexual, etc.) can find a use for Libertarian and Objectivist ideas. (If you don't know the definitions of Queer, Non-Binary, or Pansexual, Queer is an umbrella term for every non-straight identity, a Non-Binary is someone who switches back and forth between male identity and female identity, and Pansexual is a sexual orientation that is attracted to all people, of every gender and every identity, without exception.)