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The surface temp of the Sun will go up a bit as it expands but not so dramatically that it will alter its spectral class too much. The main problem is whilst the core is filling up with helium ash, the outer layers of the Sun have to expand to accommodate the heating up of the core (as the ash builds up, the core slowly contracts and heats up). So, the Sun slowly increases its luminosity, hence we get more sunlight and it heats up here (and elsewhere in the Solar System).
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