The tyres cool down after a few laps due to tyre wear - the tread thickness reduces, so becomes less insulating. In other words, the tread can't hold as much heat, so the tyres cool down quicker.
For longer races, it's fairly common practise to let the tyres overheat a bit for the first few laps, till they wear enough to cool down a bit. They usually manage to keep the temperature fairly constant after that, providing you're driving fast enough and the pressures + camber are correct.
As they get very thin, they start loosing heat even faster and then get too cold.
Regarding F1 this year, in most races the harder compound tyres are perfectly capable of lasting the whole race. They've often been described as being able to 'run all day'.
However, the recent races where they've degraded too quickly is mostly down to them overheating - either because the track is so slippery they're sliding all over the place and generating a lot of heat from friction, or because the ambient temperature is very hot.
The reason F1 tyres (and most racing tyres) degrade so much is because as they get very hot, the rubber starts to melt and they start graining very quickly.
Unfortunately, graining isn't simulated (yet?) in LFS, so the tyres don't compare very well to certain racing series (sprint tyres) or conditions.