  We have used a combination of ultrafast coherent phonon spectroscopy,
ultrafast thermometry, and time-dependent Landau theory to study the inversion
symmetry breaking phase transition at $T_c = 200$ K in the strongly spin-orbit
coupled correlated metal Cd$_2$Re$_2$O$_7$. We establish that the structural
distortion at $T_c$ is a secondary effect through the absence of any softening
of its associated phonon mode, which supports a purely electronically driven
mechanism. However, the phonon lifetime exhibits an anomalously strong
temperature dependence that decreases linearly to zero near $T_c$. We show that
this behavior naturally explains the spurious appearance of phonon softening in
previous Raman spectroscopy experiments and should be a prevalent feature of
correlated electron systems with linearly coupled order parameters.
