The functional derivative of the superconducting critical temperature $T_c$ with respect to the Eliashberg electron-phonon spectral function $\delta T_c/\delta {\alpha }^{2}F\left(\omega \right)$ is commonly used as a diagnostic tool to identify the phonon-frequency regions that most effectively enhance $T_c$. In this work, the explicit effect of temperature on the calculation of $\delta T_c/\delta {\alpha }^{2}F\left(\omega \right)$ is investigated within the isotropic Eliashberg formalism for the conventional high-$T_c$ superconductors ${\mathrm{H}}_3\mathrm{S}$ and ${\mathrm{D}}_3\mathrm{S}$ over the pressure ranges in which their $T_c$ have been measured. The results show that the functional derivative is strongly temperature dependent and, for every pressure in both systems, exhibits a robust local minimum in the $\omega \text{−}T$ landscape, characterized by $({\omega }_L,T_L)$. This characteristic point corresponds to a thermal reference condition where $T_c$ is least sensitive to redistributions of spectral weight in ${\alpha }^{2}F\left(\omega \right)$. The extracted ${\omega }_L$ correlates approximately linearly with ${\omega }_{ln}$, while the reduced ratio ${\omega }_L/\left(k_B{T}_L\right)$ correlates with ${\lambda }^{-1}$ and clusters around 7, placing $({\omega }_L,T_L)$ close to the canonical optimal-frequency scale ${\omega }_{\mathrm{opt}}\sim 7$–$8k_B{T}_c$. Motivated by these scalings, an empirical relation (mapping) is proposed that establishes a correlation between the experimental critical temperature $T_c$ and characteristic quantities extracted from the temperature-dependent functional response $({\omega }_L,T_L,\lambda )$. Over the explored pressure ranges, the resulting estimates reproduce the experimental $T_c$ values with absolute errors of 0.1–4.9 K (mean absolute error of 2.56 K). While the mapping is empirical and has been validated only within the closely related ${\mathrm{H}}_3\mathrm{S}/{\mathrm{D}}_3\mathrm{S}$ family and an isotropic treatment, the systematic behavior with pressure and isotopic substitution suggests that the identified local minimum is a robust feature of the functional derivative, motivating broader tests in other electron-phonon superconductors, including anisotropic and multiband systems.

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• Superconductivity
• Hydrides
• Eliashberg theory