Nitrogen in race tires has zero to do with maintaining a particular tire temp.
Nitrogen is used very simply because it is more consistant than air (mostly due to the lack of moisture) in the expansion rate as the tire heats up. This means that a racer will get a more consistant hot tire pressure, which helps in keeping the car handling consistantly.
Tire temps are taken, partly to ensure that the tires aren't overheating (which is a gross error state) but the real value, is measuring temps across the tire, and determining how the tire is being used. Tire temperature profiles are key in setting up a race car properly. Tire temps can teach you all sorts of things about where to set camber, toe, shock settings, etc.
And fwiw, I'm posting this from my hotel in Daytona, where me and the rest of my team are for a testing session this weekend for our three Grand Am Cup Mustangs.
