tyre temperature

Does the back tyre heat up quicker than the front one? If so, how much quicker?

That depends on how you ride. Heat is generated primarily from braking and acceleration. On the track, the rear usually runs hotter than the front, which is why tire pressures are very different for track use. To heat up the tires, accelerate and brake hard. Swerving side to side is a waste of time...

For example,

on the the track you might run, 29lbs rear, 31lbs front, the goal being that since the rear runs hotter on the track, the pressures at temperature will be equal or very close front and rear.

For street, the opposite is often true, where you run higher pressures in the front, say 34 front, 36 rear. And for long hwy/freeway runs, 36/40 or so. Higher pressures help the tires to run cooler, and to reduce wear.