Obviously, I need a hot-water bottle hat

So, did you know that the worst part about having windows right above your bed is that they are poor insulation for when it's super cold outside? Yes, it's true! In fact, not-great windows can actually be quite chilling when the temperature drops below 40 degrees. And when your head and pillow is right below said windows, it actually gets cool on your head while you're sleeping. I've done research. At this point, I can say pretty definitively, that this is not comfortable.
Last night, I slept with my sweatshirt on, with the hood on my head. I think I may have to do that again tonight, because just right now, sitting with my back to the window, I'm chilly.
These are things you don't notice when it's summer and the only problem is cooling down, and there's a fan in the room. In winter though, if you haven't managed to get really warm at all during the day, crawling into a cozy bed should be a welcome respite. But not so much when you know your head will remain cold.
Grr. I growl at this problem. Would a beanie cap be a better thing to sleep in than the hood? I'm kinda thinking yes, but I wonder if the hat would stay on well. And would it be claustrophobic to have it on at all?
This is stupid. But the issue is a real one. A complex one. I'm having a problem, people, and I need to fix it. Stat. I can't have a cold head when I sleep. That's just wrong.