The answer is “Not so much”. Your heart does need to work a little harder to keep your muscles oxygenated in a hot room and to pump extra blood to your skin’s surface, so you do breathe a little harder, but the amount of extra calories burned is relatively modest compared to other types of exercise. One study showed that women kept in a respiration chamber at 81F burned about 10 calories more an hour than women kept in the chamber at 71.6F. This also explains why you can’t get skinny sitting in a sauna. You can lose temporary water weight, but you simply aren’t burning a significant amount of calories.