I'm surprised nobody has mentioned this. Tires generally do not follow the regular friction relation. When tires heat up, they get sticky (very slightly). This is why dragsters burn out before a run, and also why you see race cars weave back and forth when there is a caution on the track and they must slow down.
Since the tire is sticky, the force due to stickiness would be a function of the area of the tire touching the road. So in effect, the total "adhesive" force of a tire to the road is:
F = friction + stickiness
where the stickiness is something like C*A where C itself is a function of tempurature and A is the area of contact.
Of course you don't run on flat tires though since there is also a rolling resistance when a tire is not perfectly rigid. Thus the big game about adjusting tire pressures to get the "best" performance you can get..