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The earth's core is about 5500C and is mostly composed of iron and nickel, probably. Presumably, it would shrink tremendously going from 5500C to 0C so in theory you could calculate the rate of shrinkage using iron's rate of thermal expansion. However the core is also under immense pressure which makes iron much denser (smaller) than on the surface of the earth. The immense temperature and pressure is a result of the action of gravity pulling the core onto itself.
The short answer I think is the earth cannot exist as we know it at anything below its core temp of 5500. Suppose we waved a magical wand that set it's temperature to 0, it would implode on itself (along with the rest of the planet) and heat right back up to its current core temp of 5500 before you could measure the effects of thermal expansion.