There's a little bit of smoke-and-mirrors going on here. Yes, the same effect happens on/in the sun (in fact it's even more extreme because the sun's gravitational field is much stronger than earth's). But no, it's not quite correct to say that the center of the sun -- or of the earth for that matter -- is "older" than the surface because the material in both bodies moves around. If you put a clock at the center and another synchronized clock at the surface and left them running for a few billion years and then compared them, you'd see a difference. But of course there are no such clocks. The only "clock" is the material that exists in both locations, and in all cases this material moves around over time. On earth there is some stability because heavier material sinks towards the center and more or less stays there, while lighter material floats to the surface and again more or less stays there. But the sun is made almost entirely of hydrogen and helium and that material is constantly being moved around due to convection.