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The only reason that the subsurface rock is cooler than magma is being able to conduct heat through to a cooler surface. When that surface becomes exterminatingly hot and remains so for a while the subsurface will be heating up too. Since the sun is estimated to remain a red giant for about one Gy it will become hotter than the surface.

1 Gy ago, life wasn’t multicellular.

I don’t think we have the ability to meaningfully predict such scales.

At that scale of time, it's not implausible that we have the technology to relocate Earth itself by then (e.g. increase its orbit).

Completely in line with our propensity for destabilizing harmonic systems in nature. Imagine we do that and all of the other planetary orbits degenerate or become dangerous.