  The family of the superconducting quasi-skutterudites
(Ca$_x$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ features a structural quantum critical
point at $x_c=0.9$, around which a dome-shaped variation of the superconducting
transition temperature $T_c$ is found. Using specific heat, we probe the normal
and the superconducting states of the entire series straddling the quantum
critical point. Our analysis indicates a significant lowering of the effective
Debye temperature on approaching $x_c$, which we interpret as a result of
phonon softening accompanying the structural instability. Furthermore, a
remarkably large enhancement of $2\Delta/k_BT_c$ and $\Delta C/\gamma T_c$
beyond the Bardeen-Cooper-Schrieffer (BCS) values is found in the vicinity of
the structural quantum critical point. The phase diagram of
(Ca$_{x}$Sr$_{1-x}$)$_3$Rh$_4$Sn$_{13}$ thus provides a model system to study
the interplay between structural quantum criticality and strong electron-phonon
coupling superconductivity.
