The average particle energy of a Fermi-Dirac gas, with zero chemical potential, is about 3.15T, where T is the temperture of this gas. To get the average energy, one needs to do an integration. The integrand is something like
\frac{x^3}{e^{x/k_BT}+1}.
I could get the result numerically. But...
Although vacuum does not mean containing nothing, there should be no preferred coordinate system in a vacuum. But if its stress-energy tensor (T) is not Lorentz-invariant, then there would be one. Therefore, T must be Lorentz-invariant.
It turns out that we have only one choice for such a...