I'm looking for a solution to vent heat from a small closet into a larger conditioned space, possibly a thermostatically-controlled in-wall fan, or ceiling exhaust fan, etc. We have one small area of the house which is currently "unconditioned" in that it was built without any ducts/registers. It's roughly 4 feet by 8 feet and 9-feet tall. It does not have any external walls, but there's an opening on one side into a home theater (an underground space which always stays cool) and a door/wall facing a large basement play area on the other side.
There's a lot of audio/video equipment in there which generates a fair amount of heat. If I leave the door closed it will quickly get to ~82 degrees in there, even when the rest of the basement is ambient at 70.
I'd like to use something to push heat from the top of the closet space out into the main basement area, and passively drawing cool air from the theater into the resulting vacuum. Is that how you would recommend addressing the situation, and do you have any ideas on how to do it?