  We study thermalization and thermal transport in single crystals of
CsV$_3$Sb$_5$ through the CDW transition by directly measuring thermal
diffusivity ($D$), thermal conductivity ($\kappa$), resistivity ($\rho$), and
specific heat ($c$). Commensurate with previous reports, we observe a sharp,
narrow anomaly in specific heat associated with a first order transition that
results in a CDW state below $\sim94$ K. While a corresponding sharp anomaly in
thermal diffusivity is also observed, resistivity and thermal conductivity only
exhibit small steps at the transition, where the feature is sharp for
resistivity and broader for thermal conductivity. Scrutinizing the thermal
Einstein relation $\kappa=cD$, we find that this relation is satisfied in the
entire temperature range, except in a narrow range around the transition. The
Wiedemann-Franz law seems to work outside the critical region as well. Below
the transition and persisting below the two-phase regime we find strong
resemblance between the resistivity anomaly and the specific heat, which may
point to a secondary electronic order parameter that emerges continuously below
the transition.

Strongly Correlated Electrons