High-temperature thin-film superconducting devices are becoming commonplace for applications such as Josephson junctions, superconducting quantum interference devices, interconnects and radiation bolometers. The thermal stability of these devices is critical and for effective hear transfer it is required to know the thermal boundary resistance at the interface of the high-Tc thin films and their substrates. Many measurements of thermal contact resistance for bulk samples and metallic thin films have been reported, but only a few measurements have been made for high-T-c films. In the present study, the thermal boundary resistance between Er-Ba-Cu-O films on MgO and SrTiO3 substrates is measured as a function of both temperature and the applied heat flux, and for the first time at temperatures as low as 19 K. The boundary resistance shows a decrease with increasing heat flux. Possible explanations for this trend are given by three proposed hypotheses.