  ErTe$_3$ is studied as a model system to explore thermal transport in a
layered charge density wave (CDW) material. We present data from thermal
diffusivity, resistivity, and specific heat measurements: There is a sharp
decrease in thermal conductivity both parallel and perpendicular to the primary
CDW at the CDW transition temperature. At the same time, the resistivity
changes more gradually. Correspondingly, while well above and below $T_c$, a
consistent description of the thermal transport applies with essentially
independent electron and phonon contributions (estimated using the Wiedemann
Franz law), in the critical regime no such description is possible; the
observed behavior corresponds to a strongly coupled electron-phonon critical
`soup.'
