There's no law that says you can't change the colour of your car, and if it makes you feel more comfortable, keep the receipt from the paint shop that did it, in the glovebox.
As for white being cooler than black, it sure is, but it still gets hot.
The main advantage of the white is how much less time it takes for the AC to cool it off.
Personally, I wouldn't do it, as the price is prohibitive, but if I planned to paint the car anyhow, then I would consider it.
A colour change paint job, does cost more than same colour.