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Keep the chemical potential at its zero-temperature value $\mu =E_F$ and put $E^{\prime }=E-E_F$. Away from $E^{\prime }=0$,Consequentlywhere

The bracket is an odd function localized to $|E^{\prime }|\lesssim k_{B}T$. When $\beta E_F\gg 1$, extending the lower limit to $-\infty$ makes the integral vanish exactly. The omitted tail is exponentially small; indeed direct integration givesThis is the low-temperature particle-number cancellation for constant density of states.

For the energy,At low temperature the first term is exponentially small. In the second, extend the lower limit to $-\infty$ and set $z=\beta E^{\prime }/2$:The remaining integral is a finite positive constant, proving the quadratic low-temperature energy correction for constant density of states: