 Adiabat with linear heat capacity

For a fixed-composition ideal gas, the reversible adiabatic process relation $c_{p}dT/T=Rdp/p$ gives $A\ln (T/T_s)+B(T-T_s)=R\ln (p/p_s)$. Under hydrostatic equilibrium, $gz=A(T_s-T)+B(T_s^2-T^2)/2$. Increasing positive $B$ at fixed surface conditions increases temperature at a given smaller pressure and raises the model's formal zero-temperature height. That finite-height mathematical boundary is not a physical absolute-zero prediction.