The thermal equation of state p=w e, where w is the ratio of the pressure, p, to the internal energy density, e, would identify it as the latter. If that were the case w would vary between 1/3 for a photon gas to 2/3 of an ideal gas. Yet, the ratio that gives thermodynamic gravitational equilibrium varies from 10^{-6} for normal relativistic stars to 10^{-4} for white dwarfs to 1 for black holes, all being equal to the ratio of the Schwarzschild radius to the radius of the star. Thus the correct density would be the rest mass density, and the cosmological pressure, p, would be negligible under most circumstances.
For the time being, let me add only that, since pressure is an energy density, the ratio w in the equation of state must indeed be a dimensionless number.
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