Most likely yes. Class A amps produce heat regardless if a signal is passing through and in many cases the current (and heat) at idle is greater than the full volume current going to the speakers.
Stupid class A question
I have a Sugden a21 integrated amp in my second system (my first class A) and I love the sound but I don't play it much in the summer due to the heat. Since it can be used as a preamp, if I get an amplifier, will the Sugden still produce as much heat if it's being used as a preamp? Thanks in advance.