But I wonder if that's such a good idea. If someone isusing the system and switches to an audio source where the tirgger is off, and then back again to a video source, you could cause problems by switching the projector on and off without time to cool. Not a good idea.
More importantly, I suspect that you might not be able to do this anyway jus twith a hard trigger on/off for a power relay since most projectors do not have a "hard" on/off, but rather operate from IR or serial commands.