We observe and comprehend the dynamical Coulomb blockade suppression of the electrical conductance across an electronic quantum channel submitted to a temperature difference. A broadly tunable, spin-polarized Ga(Al)As quantum channel is connected on-chip, through a micron-scale metallic node, to a linear $RC$ circuit. The latter is made up of the nodes geometrical capacitance $C$ in parallel with an adjustable resistance $Rin {1/2,1/3,1/4}times h/e^2$ formed by 2--4 quantum Hall channels. The system is characterized by three temperatures: a temperature of the electrons in the large electrodes ($T$) and in the node ($T_mathrm{node}$), and a temperature of the electromagnetic modes of the $RC$ circuit ($T_mathrm{env}$). The temperature in the node is selectively increased by local Joule dissipation, and characterized from current fluctuations. For a quantum channel in the tunnel regime, a close match is found between conductance measurements and tunnel dynamical Coulomb blockade theory. In the opposite near ballistic regime, we develop a theory that accounts for different electronic and electromagnetic bath temperatures, again in very good agreement with experimental data. Beyond these regimes, for an arbitrary quantum channel set in the far out-of-equilibrium situation where the temperature in the node significantly exceeds the one in the large electrodes, the equilibrium (uniform temperature) prediction for the conductance is recovered, albeit at a rescaled temperature $alpha T_mathrm{node}$.
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