It doesn't seem like the Earth's center is cooling very much if at all. I base that on a couple of scientific articles I've read. So if that's true, why isn't it cooling? What maintains the temperature? I don't think it is a fusion process like the Sun, so why isn't it losing heat? At what if it did start losing heat, what would be the effect on our climate?
Of course the Earth's core is cooling. It was heated from the original formation of the Earth and has been slowly cooling ever since as it gives off heat to the mantle, abated slightly by additional heat constantly injected back into it by the radioactive decay of some of the metals within it which offsets most of the cooling leaving the core today at a thermal equilibrium between heat out and heat back into it. But at the surface, the Earth radiates about 500 watts per square meter, but most of that energy comes from the Sun. The core makes a minor contribution, but since the actual temperature of the core is unknown, an exact value is impossible. But eventually someday, the core will grow cold.