Don Foreman writes

Power flow is pressure * volume flow, volume here being actual volume
at that pressure.

No, power = pressure * mass flow, which is why that definition of
"actual volume" instead of "free air volume" isn't useful for such
calculations.

Thermodynamics enters in if heat is lost during compression as is
generally the case.
Thus Ingersoll's generally-true note that
compressing to pressure higher than needed is inefficient.

Losing heat isn't the reason for inefficiency of excess pressure. If it
were, insulation would cure it. Regulating down is inherently
inefficient.

The loss
is at the compressor, not at the regulator. The compressor gets hot
but the regulator does not, right?

The regulator can get downright chilly. Temperature is not heat or a
measure of heat. Something can get colder and gain heat at the same
time.