The game isn't affiliated with Star Wars: The Old Republic, though it is a response of sorts to the recent debacle concerning a 'gay planet.'
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.
The premise:
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.