The sensitivity of superconducting electronics operated in the sub-Kelvin temperature range is usually limited by the hot-electron effect. Here, an increased thermal resistance due to a weakened electron–phonon coupling leads to a higher temperature of the electrons in the thin-film shunt resistors of the Josephson junctions. Cooling fins can be attached to weaken this effect. We characterized different configurations of resistors in PdAu with or without attached cooling fins by dissipating power and determining the effective electron temperature. This was done by directly measuring the Johnson noise with a SQUID amplifier. The results are compared to theory and numerical calculations on the electronic heat transport. The latter turns out to be a useful tool for the optimization of the thermal design of superconducting electronics.