  Pb$_{0.77}$Sn$_{0.23}$Se is a novel alloy of two promising thermoelectric
materials PbSe and SnSe that exhibits a temperature dependent band inversion
below 300 K. Recent work has shown that this band inversion also coincides with
a trivial to nontrivial topological phase transition. To understand how the
properties critical to thermoelectric efficiency are affected by the band
inversion, we measured the broadband optical response of
Pb$_{0.77}$Sn$_{0.23}$Se as a function of temperature. We find clear optical
evidence of the band inversion at $160\pm15$ K, and use the extended Drude
model to accurately determine a $T^{3/2}$ dependence of the bulk carrier
lifetime, associated with electron-acoustic phonon scattering. Due to the high
bulk carrier doping level, no discriminating signatures of the topological
surface states are found, although their presence cannot be excluded from our
data.
