When is the Assumption that Enthalpy is Constant over Temperature Change Invalid?
The $G{-}H$ equation describes the effect of $T$ on $\Delta G$ at constant pressure. But, along the fusion contour of $T$ vs $p$, $T$ is a unique function of $p$. So, along this contour, $p$ is changing. So the $G{-}H$ equation is not valid to use for a change of phase such as fusion. The $G{-}H$ equation was developed to describe the effect of $T$ on $\Delta G$ for chemical reactions at constant pressure, usually for the change from separate pure reactants to separate pure products at the standard pressure of $1\ \mathrm{bar}$.