Previous work using multibeam sonar to map diffuse hydrothermal flows is extended to estimate the heat output of diffuse flows. In the first step toward inversion, temperature statistics are obtained from sonar data and compared to thermistor data in order to set the value of an empirical constant. Finally, a simple model is used to obtain heat-flux density from sonar-derived temperature statistics. The method is applied to data from the Cabled Observatory Vent Imaging Sonar (COVIS) deployed on the Ocean Observatories Initiative's Regional Cabled Array at the ASHES vent field on Axial Seamount. Inversion results are presented as maps of heat-flux density in MW/m² and as time series of heat-flux density averaged over COVIS' field of view.