The derivation shown here assumes H and U are independant of temperature, right? But what if $H(T) = H_{0}+C_V \Delta T $?
Then $dG = C_V\, dT - SdT$, right? So $ \displaystyle \left( \frac{\partial G}{\partial T}\right)_p=C_V -S$ ?
$$\begin{align} \left(\frac{\partial (G/T)}{\partial T}\right)_p &= \frac{T(\partial G/\partial T)_p - G(\partial T/\partial T)_p} {T^2} \\[8pt] &= \frac{T(C_V-S) - G(1)}{T^2} \\[8pt] &= \frac{TC_V-TS-G}{T^2} \\[8pt] &= -\frac{H_0}{T^2} \end{align}$$
as desired (since $G = H_0+C_V\Delta T - TS$).
But this way we see that the H in this equation would only be containing the constant term? Of course we insert H(T) to get thee exact G, but where in my thinking process did I go wrong? How would this derivation be made correctly, assuming that H changes with Temperature?