There are some cars where you can end up with a blue outer band while the inside is green, surely restricted ones where the tyre is supposed to withstand much more power and speed, as the Baby-R which I was practicing with yesterday

but I think there could be more.
My guess is the inside of the tyre provides heat inertia to the tyre, e.g. it slows down contact patch heating in the warm up phase and then slows down its cooling in later stages. The same could be said of the rim too (if you look closely on some cars that warms up too) though I suppose to a limited degree.
The XFG is a great car to do some testing about this, it's easy to bring the patch up to temperature and then the inside will come up as well.
NB: in LFS there's no such thing as 'cold' tyres out of the pits. They're already above ambient temperature, and an easy way to check this is to let the car sit parked along the track for some time, the tyres will cool down to about 20 °C IIRC.
This has been the source of some arguing since tyre blankets may only regarded as realistic for the faster race cars.