The kagome metal ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ provides a unique platform for studying intertwined quantum orders. Combing first-principles calculations and the Eliashberg theory of electron-phonon coupling (EPC), this work reveals axial-strain effects on superconducting properties of the kagome metal ${\mathrm{CsV}}_3{\mathrm{Sb}}_5$ via EPC modulation. We demonstrate that compressive strains along distinct crystallographic directions induce contrasting phonon-mode responses: The $c$-axis compression softens M-point phonons while hardening L-point modes, whereas $ab$-plane strain directly enhances V-V orbital hybridization. These strain-induced electronic and lattice dynamics changes lead to nonmonotonic EPC evolution and monotonic suppression of superconducting transition temperature ($T_c$). Carrier doping further modulates superconductivity through dual mechanisms: Hole doping enhances EPC (λ up to 1.65) through phonon softening and Van Hove singularity (VHS) engineering, while electron doping suppresses it. Anisotropic Migdal-Eliashberg calculations indicate highly anisotropic superconducting gaps (e.g., 76.74% anisotropy under 8% $c$-axis strain), primarily governed by V $d$-orbital electrons. Moreover, self-consistent Bogoliubov–de Gennes (BdG) equations based on Wannier functions confirm that $T_c$ decreases monotonically with increasing strain, consistent with Eliashberg-theory predictions. This study establishes axial-strain tuning strategies for optimizing $T_c$ in kagome systems, providing theoretical insights for experimental manipulation of superconducting material properties.

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• Kagome metal
• Low-temperature superconductors
• Bogoliubov-de Gennes equations
• Eliashberg theory
• First-principles calculations