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Are there any indications that turning thermostats down (or up) impacts weight?

Maintaining body temperature takes energy, so being in a cooler environment would help, but it would be minimal, I'm fairly sure. Air has a much lower specific heat than water, and I remember reading a reddit discussion about drinking ice water all day to lose weight versus room temperature water. I don't remember what the result was, but it didn't take as many calories to heat the water as you'd think.

Well a calorie is the amount of energy to raise a gram of water by one degree Celsius, so if you're drinking 2 litres of water a day and it's 0 degrees vs 20 degrees, you'll burn around 20,000 calories a day from the difference.

Note though that that's calories, and not kilocalories, which is what we generally talk about for dietary purposes. So ~20kcal a day, which would represent about a 1% change for the average person. Not a huge gain.

Also, people might just eat a spoonful of more food, if they need 20kcal more per day.

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