I found this document on Class D amplifier design:
On page 6, on the right side, it shows the equations for calculating power losses in a class D amp.
It looks like in addition to the switching losses, (which I think are constant, but I can't say for sure because I don't know what some of those variables represent) there are also the conduction losses that MarkT mentioned, which are given by:
(Rds(on) / Rload) * Pout
And, there are gate drive losses for the mosfets, which appear to also be constant so can just be considered switching losses.
Anyway, given that chart in the datasheet indicates a large drop in dissipated power as the output power is reduced, I think it's safe to say that the conduction losses are the biggest contributor here to the dissipated power, and as those do vary with the volume, a class D amp will in fact draw less current and dissipate less heat when you turn the volume down or the music is quiet.