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Doesn't the earth itself provide a significant amount of heat from the core? I'm sure I read somewhere that for something like every 10 meters down you dig, the temperature raises by 1° celcius. So maybe we'd not notice a temperature drop so quickly?
The surface would eventually freeze over. But some life would almost definitely survive deep underground and underwater, near geothermal vents not unlike those that hosted the first lifeforms on Earth. And, maybe, in some billions or trillions of years, Earth would stray near another star system, get captured by its gravity and slowly thaw out, restarting the evolution of life.
Would hydrothermal vents produce enough heat? Or would the oceans freeze over? And then would there just be thermal bubbles surrounding the vents in oceanic ice?
Even if they were to, there is still the deep biosphere
The oceans would eventually freeze over, but the deep ocean could stay liquid for tens of millions of years. Ice is a pretty good insulator, and there is more than one moon in the solar system suspected to have liquid oceans under a layer of ice.
Not sure how quick exactly, but the earth doesn't provide enough heat, not even close. Kurzgesagt has a video on a similar subject, without the ~~trillions~~ 1.7e17 Watts showering the earth every second we'd get awfully cold awfully quick. They are talking about slowly moving away from the sun, but they conclude it would get real icy