Can't say exactly why the bottom part seems cooler when you just get in bed; actually, it doesn't work that way always in my experience.
What I can say is why the bottom part is never warmer; being in permanent contact with another surface will not affect its temperature at all if the other surface isn't generating heat and is at the same temperature.
As a matter of fact, a sweater for example doesn't warm anything up; if you put your sweater to your chair, you can come back 10 hours later and find the chair as cold (again - provided it's a controlled environment with a stable room temperature and no object generating heat) than it was when u put the sweater around it. Cloth - and blankets for that matter - don't "warm" things up, just prevent heat from dissipating as quickly. Thus, two surfaces touching at the same temperature won't get warmer together.
A little off topic, but I always like to pinpoint this when I get into ANY thermodynamics conversation (I have no idea why, maybe I'm just crazy!): cold doesn't exist in reality, it's just lack of heat. 🙂