I think Discount tire charges about $25 to fill your tires with nitrogen. It does reduce the pressure change due to temperature. Simple physics and ideal gas law
Ideal gas law - Wikipedia, the free encyclopedia . The coefficient of expansion for nitrogen is less than that for oxygen. But since normal air is only 21% oxygen I can't see it being a massive difference without running the numbers.
Another article about the subject
» Nitrogen-Filled Tires & Better Gas Mileage?*on*Blueprint for Financial Prosperity
I can't speak to running cooler, but I would tend to doubt that unless it is somehow a function of running closer to proper pressure all the time.
I recall from my days in the Navy that the tires on our aircraft were filled with dry nitrogen.
Robert