Do you have a dumb question that you're kind of embarrassed to ask in the main thread? Is there something you're just not sure about?
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Notes -
From first principles, active bodily processes to either heat or cool one's flesh likely consume calories. Temperature gradients need to be maintained. In fact, I've seen work posing the question of the effect of indoor environment control on caloric expenditure (if you have electricity doing the work of regulating your environment, you likely have to expend less). Given that most people maintain a body temperature above that of ambient, it is theoretically plausible (even likely) that increased body temperature would increase caloric expenditure.
Of course, the rub usually comes in terms of magnitudes. How big is the effect? A casual scan of the literature doesn't turn up anything all that great. So, I would maintain my personal belief that there is likely a positive effect, but extremely low confidence in any sense of an estimate for magnitude.
Concerning equations, the question always is what it is that you're trying to do. For very small groups, you can go through a very intensive process of measuring all sorts of body characteristics, down to the size of individual organs, and use some pretty detailed estimates to try to get really close. Most people don't do anything like that; it's just too much effort. Instead, people often want to collapse larger-group data into a handful of variables for ease of estimation, knowing that any such effort will inherently have variability and error bars. The equation that you get, and how much variability it has, depends on which type of population you're targeting and which variables you're trying to collapse it down to. Obviously, targeting larger/smaller population types tends to increase/decrease variability; similarly, increasing/decreasing the number of variables (under mild assumptions of them being correlated at all to the dependent variable and not entirely codependent) tends to decrease/increase variability. If you're considering fit and athletically-active populations, a lot of folks recommend the Cunningham equation. It also does not include typical body temperature. I'm not sure what the codependence will look like, what the rough magnitude of the effect will be, and how much additional variability you could cut out by including typical body temperature, just because I'm not aware of anyone who has taken the time and money to specifically explore it.
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