Hospitals are right, and there are additional factors. Most pathogens really like human body temperature, so keeping the difference high enough between rooms and bodies helps cutting their growth rate (and it’s a quadratic correlation, not linear). Also, lower room temperatures have shown to be preferable to reduce patient mortality in general - The body doesn’t have to do tons of work to cool you down, which is additional strain on the circulatory system and reduces available water due to sweating. It’s no coincidence that mortality rates rise dramatically during heat waves.