  At zero temperature, a Josephson junction coupled to an ohmic environment
displays a quantum phase transition between superconducting and insulating
phases, depending whether the resistance of the environment is below or above
the resistance quantum. At finite temperature, this so-called Schmid transition
turns into a crossover. We determine the conditions under which the temperature
dependence of the thermal conductance, which characterizes heat flow from a hot
to cold resistor across the Josephson junction, displays universal scaling
characteristic of the Schmid transition. We also discuss conditions for heat
rectification to happen in the circuit. Our work can serve as a guide for
identifying signatures of the Schmid transition in heat transport experiments.
