From Al-Faitha Foundation, a support organization for GLBT Muslims [emphasis added]:

During the time of the Prophet Muhammad (d. 632 AD) there was not one single case of a reported punishment or execution for homosexuality or same-sex acts. The first execution to ever have been carried out was during the time of the third Caliph, Omar, who ordered a homosexual man to be burned alive.

Scholars at the time differed in opinion on this form of punishment, arguing that no human should be burned (according to the traditions of the Prophet Muhammad), thus it was decided that homosexuals should be thrown off the highest building and then stoned to death.

Baby steps, I suppose…

In other news, the universe will be turning 6008 years old soon! Happy birthday, Universe!