recommendations for amplifiers that don't emit tons of heat

the biggest power consumer in a tube amp is usually the heaters for the cathode

This statement is rarely true- I think the exception used in audio is the 6C33. For most power tubes - like a KT88- the heat made by the cathode is insignificant compared to the heat made by the tube as a result of plate current though it (its class of operation).

As a result, compared to a solid state amp of the amp power and **same class of operation** , tube amps will make marginally more heat.

If you put a tube amp in standby (no B+) even after sitting there all day the tubes won't burn you (the exception being the 6C33). But after being on only a couple of minutes the same tubes will be much hotter if B+ is also applied!

The problem is that if a solid state amp can't bring home the bacon, then it really doesn't matter how much heat it makes. In high end audio, the **sound** produced is the goal. 

Y'know, if you like the sound you are getting there is a much less expensive option, which is simply to have a vent installed to the outside that allows the heat to escape the room rather than warming it up. This can be done for a few hundred dollars at most (which if really sophisticated might include a wall switch with a fan to help move the air). I've seen this be very effective!