 Adiabatic scaling of a Fermi gas with power-law density of states

For density of states $AV{ϵ}^a$, $a>-1$, the grand canonical ensemble has $\log \mathcal{Z}=V{T}^{a+1}f(\mu /T)$. Strict monotonicity of Fermi gas entropy per particle makes $\mu /T$ constant at fixed entropy and particle number. Therefore $V{T}^{a+1}$ and $P{V}^{(a+2)/(a+1)}$ are constant along a reversible adiabatic process.