I'm not sure I understand, though, why Class A would run cooler when outputting signal into a load. It's tempting to think that could result from a portion of the current that would otherwise flow through some of the output transistors (at zero signal conditions), being diverted through the speaker instead.
Yes that is the reason. At no load (or I meant to say no input signal) you are actually running at 50% current output and all that current has to be dissipated internally in the amp somewhere, across the bias circuit if you will (it is the high bias that makes the current flow at 50% when at zero input signal). When you play music the +ve signal requires more current (but it all dissipates in the speaker) and on the -ve cycle it actually diverts current from the bias circuit to the speaker (meaning the amp generates less heat).
The amp may use more power from the AC but it will likely run a bit cooler - no enormous difference and I gues it may vary a bit on the exact topology and type of output devices.