From looking at the TAS around 1 second before he hits the boss chamber the guy paused for about 50 frames and then unpaused the game. I'll show you this from IRC as well:
<Inuyasha> Quartz: The game keeps track of the time the boss's first cycle should start (there's a delay when you first enter), however they neglected to make this timer stop when you pause the game
<Cruizer> yeah quartz, excerpt from the tas:
<Quartz> That's the only thing I get!
<Inuyasha> so you pause it, the timer still counts down, and when you reach the boss the cycle starts immediately, from what I gather
<Cruizer> "!Final Zone Pausing in the beginning will increase the counter for when the boss fight will start, while not costing any ingame time."
<Quartz> Oh.
<Cruizer> not immediately, at a different time
Hope that helps.