You are quite right about TRUE Class A. It does run hottest on no/low signals.
Some here have mentioned modern Krells, by which I mean post around 1990. These do not run particularly hot on no/low signals. How can this be? it is because they are not TRUE Class A, whatever Krell may claim. They run in Class B and have so-called anticipator circuits which sense an increased signal coming in and re-bias the amp to Class A and allow it to morph back to Class B when the volume has died down.
I have never discovered any scientific basis for the claim that this re-biasing can occur so quickly after detecting the increased signal that it can power the sound of that signal. It seems impossible since the cable runs in the amp are relatively short and the signal is travelling in them at close to the speed of light. A myth put about by Krell so they could play in Green Park while still claiming to produce Class A amps. Yes they will exhibit patents but anyone can get a patent. The issue is: is the move to Class A completed before the detected incoming signal is amplified? The patent makes no claim that it is.
No guys. A TRUE PURE Class A amp like my pair of late 1980s Krell KRS200s, uprated by Krell to 400w per channel runs HOT. And as Jason says, there aren't many of them.