Various human heat stress indices have been developed to relate atmospheric measures of extreme heat to human health impacts, but the usefulness of different indices across various health impacts and in different populations is poorly understood. This paper determines which heat stress indices best fit hospital admissions for sets of cardiovascular, respiratory, and renal diseases across five Australian cities. We hypothesized that the best indices would be largely dependent on location. We fit parent models to these counts in the summers (November-March) between 2001 and 2013 using negative binomial regression. We then added 15 heat stress indices to these models, ranking their goodness of fit using the Akaike information criterion. Admissions for each health outcome were nearly always higher in hot or humid conditions. Contrary to our hypothesis that location would determine the best-fitting heat stress index, we found that the best indices were related largely by health outcome of interest, rather than location as hypothesized. In particular, heatwave and temperature indices had the best fit to cardiovascular admissions, humidity indices had the best fit to respiratory admissions, and combined heat-humidity indices had the best fit to renal admissions. With a few exceptions, the results were similar across all five cities. The best-fitting heat stress indices appear to be useful across several Australian cities with differing climates, but they may have varying usefulness depending on the outcome of interest. These findings suggest that future research on heat and health impacts, and in particular hospital demand modeling, could better reflect reality if it avoided "all-cause" health outcomes and used heat stress indices appropriate to specific diseases and disease groups.