Green’s Dictionary of Slang

(gay) the final hour before a bar closes when one becomes less selective in the choice of a sexual partner.

1963		J. Rechy City of Night 187: He shrugged his skinny shoulders, glancing nervously around. It’s almost desperate hour and he hasnt made a conquest for tonight.
c.1972		‘H. Smith’ 3 Way — Only Way 1115: When they pulled up in front of the [leather bar], it was ten minutes to two. ‘Desperate hour,’ Stanley said. ‘There should be a lot of people in there’ [Simes:DLSS].
1984		R. Stevenson On other Hand Death 77: ‘I wasn’t really keeping track of the time last night. Anyway, not until desperation hour rolled around’ [Simes:DLSS].
2003		K. Cage Gayle 65/2: desperation hour n. last hour before the bar closes and one still hasn’t picked up a sex partner for the night (also: DH).