In narrow YBa₂Cu₃O₇ films, we have measured the delay time between the excitation by a critical current pulse and the voltage response, and analysed it on the basis of a time-dependent Ginzburg-Landau equation to provide the gap relaxation time. In the present experimental situation, this parameter can be identified with the phonon escape time, and so it provides information on the film-to-substrate thermal resistance. Our results for various conditions of thickness and of temperature are consistent with other independent data found either in our laboratory or reported in the literature. This new procedure is fast and applicable at any temperature below T_c.