Air temperature doesn't have much to do with it.
Try running up the Brecon Beacons in shorts and a T shirt with the snow flurrying around you.
Muscle core temperature however, does have a lot to do with it. It takes a while for the cardiovascular system to get oxygen to the muscle fibres.
It has been said often "warm up properly". This means 'stabilise the energy transportation system' within your body.
On the inverse, it takes 10 - 20 minutes of non-activity to get the muscle core temperature down AFTER a demanding exercise session.
Don't just jump into the shower, when you get out you will still be sweating. Sit down for 20 mins ( and have a dump ), and then have a shower.