I don't know, looking at this obesity map[1] I don't see much of a difference between northern/southern states. If anything there are more southern states with extreme (35%+) obesity. Doesn't mean it doesn't contribute partially though. It's also very crude data.

[1]: https://stateofobesity.org/adult-obesity/

Here's a graph with the prevalence of AC by region. It could also be related to the affordability of heat (my grandparents kept the house very cool). I have no idea if there's something to this, but interesting to poke at.

https://www.eia.gov/consumption/residential/reports/images/a...

I'm middle-aged, and even out in the sticks we had window air-conditioning. And affordable indoor heating has been a thing since long before I was born. Yet oddly enough, previous generations weren't butterballs. I didn't play a lot of Nintendo growing up, though.

An interesting hypothesis, but I think it easier to point to sedentary lifestyles and poor Western diets before I go off on a tangent like that.

How does that explain the study showing weight gain in labratory animals with controlled diets?

Interesting. I never heat or cool my house (other than opening the windows), and while not being particularly careful with my diet, I've never had any weight issues. I think your hypothesis merits further investigation.

I would guess it's more like environmental temperature leads to behavioral changes that then affect the average energy expenditure.