Finite temperature thermal corrections can be obtained by starting with the expansion derived earlier: Note that
The term proportional to is a small thermal correction to the T=0 limit. As such, it is small and we can replace the appearing there with to the same order in T:
Solving this, now, for (which is equivalent to solving for ) gives
where the second line is obtained by expanding about x=0.
In order to obtain the thermal corrections, one must expand the average occupation number formula about the value using the expansion obtained above for and the do the integrals. The result is simply
The thermal correction is necessary in order to obtain the heat capacity at constant volume, which is given by
Using the above expression for the energy, one finds
From the thermally-corrected expression for the energy, the pressure can be obtained immediately: