This step (θ u/θ s) t = (θ u/θ s) v+(θ u/θ v) s (θ v/θ s) t with subscript t is possible, which shows that we just want to investigate the internal energy change of entropy change at constant temperature. Theoretically, you can change this step to p and v, but the content of your partial derivative is different. Because there are many changes in thermodynamics, different results will be obtained by choosing different states.

I just started to learn thermodynamics, and I can't help you more.