  We present a study of the lattice dynamical properties of superconducting
SrPt$_3$P ($T_c = 8.4$ K) via high-resolution inelastic x-ray scattering (IXS)
and ab initio calculations. Density functional perturbation theory including
spin-orbit coupling (SOC) results in enhanced electron-phonon coupling (EPC)
for the optic phonon modes originating from the Pt(I) atoms, with energies
$\sim 5$ meV, resulting in a large EPC constant $\lambda \sim 2$. An overall
softening of the IXS powder spectra occurs from room to low temperatures,
consistent with the predicted strong EPC and with recent specific-heat
experiments ($2\Delta_0 / k_{\mathrm{B}}T_c \sim 5$). The low-lying phonon
modes observed in the experiments are approximately 1.5 meV harder than the
corresponding calculated phonon branch. Moreover, we do not find any changes in
the spectra upon entering the superconducting phase. We conclude that current
theoretical calculations underestimate the energy of the lowest band of phonon
modes indicating that the coupling of these modes to the electronic subsystem
is overestimated.
