  The quasi-skutterudite superconductors $A_3T_4$Sn$_{13}$ ($A$=Sr, Ca; $T$=Ir,
Rh, Co) are highly tunable featuring a structural quantum critical point. We
construct a temperature-lattice constant phase diagram for these isovalent
compounds, establishing Ca$_{3}$Rh$_4$Sn$_{13}$ and Ca$_{3}$Co$_4$Sn$_{13}$ as
members close to and far away from the structural quantum critical point,
respectively. Deconvolution of the lattice specific heat and the electrical
resistivity provide an approximate phonon density of states $F(\omega)$ and the
electron-phonon transport coupling function $\alpha_{tr}^2F(\omega)$ for
Ca$_{3}$Rh$_4$Sn$_{13}$ and Ca$_{3}$Co$_4$Sn$_{13}$, enabling us to investigate
the influence of the structural quantum critical point. Our results support the
scenario of phonon softening close to the structural quantum critical point,
and explain the enhancement of the coupling strength on approaching structural
instability.
