  Cross-correlated measurements of thermal noise are performed to determine the
electron temperature in nanopatterned channels of a GaAs/AlGaAs heterostructure
at 4.2 K. Two-dimensional (2D) electron reservoirs are connected via an
extended one-dimensional (1D) electron waveguide network. Hot electrons are
produced using a current $I_{\text{h}}$ in a source 2D reservoir, are
transmitted through the ballistic 1D waveguide and relax in a drain 2D
reservoir. We find that the electron temperature increase
${\Delta}T_{\text{e}}$ in the drain is proportional to the square of the
heating current $I_{\text{h}}$, as expected from Joule's law. No temperature
increase is observed in the drain when the 1D waveguide does not transmit
electrons. Therefore, we conclude that electron-phonon interaction is
negligible for heat transport between 2D reservoirs at temperatures below 4.2
K. Furthermore, mode control of the 1D electron waveguide by application of a
top-gate voltage reveals that ${\Delta}T_{\text{e}}$ is not proportional to the
number of populated subbands $N$, as previously observed in single 1D
conductors. This can be explained with the splitting of the heat flow in the 1D
waveguide network.
