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:
