Sorry, not true.

As a general rule, warmer fluids and gasses rise amongst cooler fluids and gasses, because they expand and become less dense (and therefore lighter) when heated - convection. I doubt that convection is that big a factor in how heat flows around the human body, given all the pumping, churning and general mixing of fluids going on.

I'd imagine that conduction is the dominant factor and (like radiation), it doesn't care which way "up" is.

There are exceptions to warmer fluids rising via convection, too. Between about 4 deg C and freezing water starts to expand again as it cools (and expands a whole lot more on freezing). This means that you can have cool water rising through warmer water via convection.

But, hats are good :-)

eeph