Well the reason it turns red is because it overall becomes cooler.
It doesn't go cooler
overall... or, well, the mean temperatures inside the star would go down somewhat since thelower temperature areas will be fairly much bigger in proportion than now as far as I know, but not
all portions of the star cool down. The temperatures at core will rise substantially according to my understanding.
It's just that the visible surface (and whole lot of stuff below it) will become cooler, which causes the peak of the black-body spectrum distribution of the star to move towards lower energy end (longer wavelength), which in visible spectrum means that it'll appear more red. Also, surface intensity (per unit of area) will go somewhat lower and obviously the density of the Sun will go
way down - in fact it's likely that the inner planets will keep orbiting inside the hot gas for some reasonably long time.
The power output of the star, however, will increase and thus the net brightness will increase... And what comes to the core, the temperature will rise
substantially because the helium fusion starts more and more replace hydrogen fusion, since helium fusion requires more energy to happen (what with helium nuclei having electrical charge of 2e as opposed to hydrogen nuclei's 1e).
...and please don't bother pointing out how inprecise terms "hydrogen fusion" and "helium fusion" are, I just can't bother to go through all the detail at the moment.
