We present experimental observations of picosecond Joule heating inside the gold metallization of ErAs:GaAs photoconductive switches. Femtosecond laser time-domain thermoreflectance is employed to resolve the fast thermal dynamics in the central switch electrode during generation and transmission of rf electrical pulses. Sharp features in the thermoreflectance signal scaling quadratically with the bias/pulse voltage reveal Joule heating with durations 5 ps inside the gold metal. The temporal shape and rise time of the signals is in excellent agreement with the theoretical pulse wave form and subpicosecond carrier lifetime of the active medium. Probing different locations on the electrode shows the propagation, attenuation, and dispersion of the electrical pulses along the coplanar waveguide structure. Overall, the presented photothermal experiments demonstrate great potential to reveal the internal dynamics of photoconductive switches, characterize transmission lines, and study ultrafast heating in metals.