 Temperature-linear pressure and volume-independent heat capacity

For a smooth simple compressible system, $C_V=T(\partial S/\partial T{)}_V$ and the Maxwell relation $(\partial S/\partial V{)}_T=(\partial p/\partial T{)}_V$ give $(\partial C_V/\partial V{)}_T=T({\partial }^{2}p/\partial T^2{)}_V$. Thus an equation of state $p=a(V)T+b(V)$ implies that $C_V$ depends on temperature alone within a smooth phase. It does not require $C_V$ to be temperature independent.