If you think about the way they do the show, the Stig works with the person for some time, then they run 10 laps from a stand still. I would guess that means warm tires.
Also, think about how they have to drive it. Line up the car, hit the gas, start the clock, cross the line, drive it around and line up to do it again. Same as in GT5, so what's the problem? Would you really want to wait for a load screen in between each lap you attempt?