This question is usually asked when conversations about geothermal come up here, and for good reason because it’s a valid question. The long and short of it is that the amount of thermal energy that humans would be able to realistically extract pales in comparison to the amount already lost to heat flux through the surface, the amount generated via radioactive decay and frictional forces from gravity/tidal sources and convection in the core. Some answers on a relevant stack exchange answer with more general calculations [0].

Edit: Off the top of my head, it’s estimated that total core heat is approximately 10^31 J, heat loss at a rate of 47TW, (forgive me for not carrying out the calculation completely or for sources on those numbers, they can be easily searched, I’m on mobile), so the orders of magnitude difference already are so vast it’s a rounding error in siginificant figures.

[0] https://earthscience.stackexchange.com/questions/2302/can-th...

Scale wise, look at a cross section of an apple.

We live above the skin, all the drilling we have ever done at depth has yet to pierce the skin through to the white apple interior.

Any "cooling" we do would be preemptively use the heat within the apple skin that is already slowly radiating outwards .. and the pinpricks, or tens of thousands of pinpricks we might make would not cause "widespread cooling of the apples interior".

See: (actual geoscientists): https://www.ga.gov.au/ausgeonews/ausgeonews201306/geothermal...

An astronomer/geophysicist could answer better, but my understanding is nothing we could possibly do would change the planets temperature the tiniest fraction of a degree.

I also read somewhere that if the sun disappeared and the earth flew off into space, life would persist in the oceans for millions of years because it would take that long for them to freeze to the bottom.

According to a quick web search, the internal energy potential of the Earth is 10^31 joules. Humans use about 5.8 × 10^14 annually.

That's around 170 trillion years if I did the math right.

> widespread cooling

You might want to look into just how much heat is contained within the Earth, how much escapes naturally every year, and how much is replaced by radioactive decay. Suffice it to say ..... no. There is no way that mankind can extract enough energy to make any real impact on the Earth's interior. The only potential issue here is closer to the surface, i.e. that injecting water into rock strata can trigger small quakes where there were none before (as we've found out via hydrofracking elsewhere).

Suffice it to say there are no easy solutions to the problem of baseload power. Expecting everyone to live in the dark is unrealistic. Expecting solar power to magically stop being inefficient and unworkable in most locations is unrealistic. Expecting the same of wind power likewise. So we're left with not a lot of good options, only less bad ones. That's life... and that's engineering. There are no perfect solutions... just ones you can live with.

At a planet scale no, but locally these geothermal plants can have an effect. Some Icelandic geysers for example have weakened or even stopped due to geothermal power plants in the area.

The US NPS has cited this as a reason for not installing geothermal generation near the Yellowstone hotspot. They don’t want to potentially impact Old Faithful.

At the scales were talking about it's somewhere between Going to the beach and getting shot by a swimming dog carrying a gun in its mouth and the problems caused by doing flybys of Jupiter.

https://what-if.xkcd.com/146/