Something I have been wondering: Why don't data centres use the excess heat for a sort of energy recuperation, turning at least some of it back into electricity?

Because it's not economical, the required hardware is unlikely to pay for itself during its lifetime. The gradient is too small (~50C), which means low Carnot efficiency. Additionally, extraction of low-enthalpy energy involves obstruction of heat transfer, meaning lower cooling efficiency. It may have been a different story if we had computer hardware able to efficiently operate at 200-300C.

Even steel plants which deal with significantly higher waste heat gradients rarely bother with recovering energy.

It's not anywhere near hot enough to generate steam and make electricity.

There are uses for low grade heat but they require colocation and careful design, which costs more than just dumping the heat.

The concept of waste-heat-to-power (WHP) exists, but its efficiency is limited by thermodynamics. Basically, heat energy is not equal to usable energy. All energy ultimately wants to be heat energy, and it is much easier and more efficient to go from electrical or mechanical energy to heat than vice-versa. Therefore, when you do have an application that actually wants heat, not electricity, such as a public swimming pool or district heating, it is way more efficient to use your waste heat as heat. Even in cases where the desired temperature is wildly different from that of your waste heat, you can convert one heat level into another very efficiently using heat pumps.

Undecided just did an episode on a waste heat machine that is being slowly rolled out to industry. The founder of the company is also the guy who invented the Super Soaker.

https://www.youtube.com/watch?v=MuQRxatte5g

Look up Carnot efficiency. The maximum amount of work you can theoretically extract depends only on a temperature difference. For a datacenter running chips at 100C into ambient air at 60F, it's about 25%. So even with perfect capture, you are guaranteed to lose 3/4 of your input energy to the datacenter as heat anyway.

For comparison, an IC engine has a Carnot efficiency of something like 80% on paper, but the reality you get is only 20-30%

Carnot's law of thermodynamics says you can only do this effectively if you can run the computers at a few hundred, or ideally thousand, degrees C.

Landauer's principle says that even if we could build computers to work at those temperatures, they'd need more power anyway.

I saw on TV a long time ago that a funeral home's "energy" (burning bodies) was used to heat homes somewhere in Europe.

We can just use data centers for heating too...maybe turn around all these protests against them

The degree to which you can extract energy from heat depends on the temperature difference compared to ambient. Efficient power stations all need super heated steam (like 600C). This would be like 100C max which is not very useful for generating electricity. It's fine for heating houses and swimming pools though.