Here is a Twine game by Anna Anthropy that's all about searching for an elusive gay planet. It's called, as you might have guessed, The Hunt For The Gay Planet.
You've heard rumors of a secret paradise planet where people like you can be people like you, a glittering world where women walk arm-in-arm with women, where you can feel the heat of a lady's reciprocating gaze without having to feel the burn of a thousand judgemental stares on your skin.
Lesbionica. The gay planet. Let's find it.
Like other Twine games, it plays somewhat like a choose-your-own-adventure. And, like much of Anna's work, you might consider it NSFW. (Update: full disclosure...Anthropy was housemates with me and a mutual friend in the summer of 2012.)
I'd say this runs about 15 minutes, and it made me chuckle a few times—both out of the strength of Anna's writing, and also because the idea of a 'gay planet' is so absurd/silly/crazy.