That's funny, I find it much easier to fall asleep in a cold environment. Then again, I also like to use a heavy blanket, so maybe it's the weight more than the cold that's helping me.

I think that's also consistent with the idea behind a hot shower. The shower doesn't help by increasing your body temperature, in fact it does the opposite. The hot shower induces the body to try to cool down, so near-skin blood vessels swell, and that dumps heat into the cold air, which reduces your core temperature, and a reduced core temperature helps you fall asleep.

I think where I read about this was Why We Sleep from Matthew Walker. But he suggests just washing your face with warm water, as opposed to a shower.