My understanding is that despite the higher TAS, there ISN'T actually the equivalent of any more air flowing through the engine. In 10 seconds of flight, there is the equivalent of 10 seconds of the CAS/IAS air passing through the engine.
Let's test that argument.
If we ignore the slight difference between EAS and CAS then for a constant CAS we must have a constant dynamic pressure.
Pdyn = 1/2 Rho TAS squared.
Lets call msl density Rho1
And let's call msl TAS TAS1
And let's call Pdyn Pdyn1
At msl Pdyn 1 = 1/2 x (Rho1) x (TAS1) squared
Now let's go to an altitude at which the density is 1/4 of the msl value
For the same CAS we must have the same Pdyn and for the same Pdyn we must have
Pdyn 2 = 1/2 x (1/4 Rho1) x (TAS 2) squared.
For the CAS to remain unchanged Pdyn 1 must be equal to Pdyn 2.
So to compensate for the fact that Rh0 2 is only 1/4 of Rho 1, (TAS 2)squared must be four times (TAS 1) squared.
This means that TAS 2 must be 2 x TAS 1
So in climbing to our altitude at which density is 1/4 of the msl value our TAS has doubled to maintain constant CAS.
Now let's look at what this means for the air mass flow rate
Mass flow rate of air going though the air linlet isl
Air Mass Flow = Inlet Area x Air Density x TAS
At msl we have Inlet Area x Rho1 x TAS 1
At our higher altitude we have Inlet area x 1/4 Rho1 x (2 x TAS1)
This means that the mass flow rate at the higher altitude is only 1/2 of the msl value.
This reduction in air mass flow rate causes the thrust at any given combination of CAS and RPM to decrease as we climb.