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 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 ∆T e in the drain is proportional to the square of the heating current I 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 ∆T 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.The transport properties of a one-dimensional (1D) waveguide are dominated by the wave-like character of electrons. The nanoscale confinement potential is typically created by applying advanced lithographic methods to high mobility two-dimensional electron gases (2DEGs). In ballistic 1D waveguides electric conductance quantization is observed 1,2 and is shown to scale linearly with the thermal conductance at low temperatures.3-5 This indicates the validity of the WiedemannFranz relation in the ballistic 1D regime, when electronphonon and electron-electron interactions can be neglected. 6,7 In previous works by van Houten et al. 3 and Chiatti et al.4 comparable heating measurements between two AlGaAs/GaAs 2DEGs were performed. The two 2DEGs were connected via a single quantum point contact (QPC) and the increase in electron temperature ∆T e of the indirectly heated 2D reservoir was measured by a second QPC. ∆T e was found to be proportional to the number of populated subbands N of the QPC. The question that arises is how the mode-dependent heat transfer evolves in networks of extended 1D waveguides, where phase-coherent effects have been investigated inand out-of-equilibrium. 8,9Here, we perform cross-correlated electronic noise measurements to determine the charge carrier temperature in an extended 1D waveguide network made from an AlGaAs/GaAs heterostructure. We investigate the heat transport through an asymmetric quantum ring, 9 which is a network of 1D electron waveguides with 2D contacts as depicted in Fig. 1. A global top-gate enables the control of the conductivity of the 2D reservoirs and the 1D electron waveguides. One electron reservoir is heated above the lattice temperature via the current heating a) E-mail: riha@physik.hu-berlin.de technique.10 The increase in electron temperature ∆T e of the other electron reservoir is extracted by the means of Johnson-Nyquist noise thermometr...