We investigate from first principles the temperature-dependent electron-phonon ($e$-ph) renormalization of the fundamental band gaps of diamond, silicon, and gallium phosphide, with many-electron self-energy effects included, using the $GW$ and $GW$ perturbation theory methods for the band energy and $e$-ph coupling, respectively. Our results show that the $e$-ph renormalization of band gaps is enhanced by $\sim 50\%$ at the $GW$ level compared to results obtained with density functional theory (DFT) and density functional perturbation theory for all three materials. Moreover, the temperature dependence of the band gaps predicted by $GW$ and $GW$ perturbation theory shows excellent agreement with experimental measurements, with significant improvements from results at the DFT level. Our analysis reveals nonuniform $GW$ self-energy corrections, both in the $e$-ph coupling from different phonon modes and in the phonon-induced contributions (the Fan-Migdal and Debye-Waller terms) to the electron self-energy. Our findings emphasize the crucial role of many-electron effects in the temperature-dependent $e$-ph renormalization of band gaps and highlight the necessity of including $GW$ self-energy effects going beyond standard DFT approaches for accurate descriptions of $e$-ph phenomena.

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• Electron-phonon coupling
• Electronic structure
• Phonons
• Density functional theory
• First-principles calculations
• GW method