The double pulsar PSR J0737 — 3039A / B consists of two
neutron stars in a highly relativistic orbit that displays a roughly 30 - second eclipse when pulsar A passes behind pulsar B. Describing this eclipse of pulsar A
as due to absorption occurring in the magnetosphere of pulsar B, we successfully used a simple geometric model to characterize the observed
changing eclipse morphology and to measure the relativistic precession of pulsar B's spin axis around the total orbital angular momentum.
So let's just agree to subtract it out
as completely irrelevant to a discussion of thermodynamics, unless the «air» in question is inside the core of a
star that is in the peculiar state where it is fusing oxygen and nitrogen or sometimes fissioning them with fast
neutrons (the only processes I can think of that might
change their baseline mass - energy by altering their strong nuclear interaction energy).