You are right, for standard state with ##p^0## it is ##G^0 = f(n_1,n_2,...,n_N) \neq f(p)## and
$$\left( \frac{\partial \Delta_{R}G^0}{\partial p} \right)_{T,\vec{n}}=0$$ $$\left( \frac{\partial \ln K}{\partial p} \right)_{T,\vec{n}}=0$$
Just for standard state with ##p## this derivation is not...