  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 node's geometrical capacitance $C$
in parallel with an adjustable resistance $R\in \{1/2,1/3,1/4\}\times h/e^2$
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 $\alpha T_\mathrm{node}$.
