The trick is to make good skin contact with your legs before you heat it up. You are a mammal and can easily dissipate over 100 watts, it is your human super power. The bottom will only get a bit warm as you pump away all the heat.

In days of old I used to keep the fans from coming on for an aluminum PowerBook with this trick.

You can even lose weight this way!

(I’m mostly kidding but not entirely. Most of our resting state calorie consumption is spent on thermal homeostasis, and the greater a temperature gradient your body needs to overcome whether to warm up or cool down, the more calories it’ll burn in doing so. Although if you’re sitting in the freezing cold or in an air conditioned room and putting a warm laptop on your lap, you’re probably going to be burning fewer calories since you’re nudging your body temperature in the right direction.)

I am convinced this is why children get so dead-tired after the swimming pool - even if only wading around. They have a small body mass and they need to regulate a ~20 degree temperature differential ...

That’s a logical enough deduction to file under “accept as scientifically valid unless and until proven otherwise,” at least as far as I’m concerned!

> whether to warm up or cool down, the more calories it’ll burn in doing so

To warm up yes, but to cool down? How would it make sense to generate more heat (burn calories) to cool down?

Our bodies are less efficient but no less incredible at lowering their temperature as they are at raising it. See for example the production and excretion of sweat takes energy, as does increased heart rate to improve circulation in an attempt to a) handle/average out local hot/cold spots, b) improve shedding of excess heat via radiation in the extremities, which is one of the reasons we have weird thin, fin-like protrusions sticking out of our heads on either side: pumping blood through these thin slabs with high surface area gives maximal chance to reduce the temperature of the blood as the air flows over them. Again the the same with the upper and lower extremities, which radiate heat at greater levels than the trunk. Also, increased, shallow breathing to attempt to cool down blood circulating in the high surface area exposed in the alveoli of the lungs (which is normally why exhaled breath is much warmer than the local air temperature). There are other mechanisms as well, I am by no means an expert but thermal homeostasis is of extreme importance and we’re quite good at it - and like any other physics process, heat exchange is a form of work that requires the expenditure of energy.

I remember reading a study that for male laptop users the extra heat has a small but real impact on fertility

A bonus, though a very small and unreliable one.

Better to just get the hardware upgrade. It's pretty quick and seems harmless.

Ouch, is that where the heat dissipates from?

The human heat sink. Sounds like something from The Matrix.