You're laying it on a little overly thick. Hospitals in blue states have also done the same thing. Also I think you're thinking of GOProud the gay conservative group that formed because Log Cabin are to liberal, and actually focus on gay issues. Log Cabin has endorsed no candidate.

Although I do agree that it's easy to see why people assume Republicans and conservatives are more likely to be homophobic. It is a part of the Republican platform to constitutionally ban gay marriage, and they have lead the vast majority of campaigns against gay rights.