temperature-wise, there are only two types of tyre - ordinary tyres and snow tyres.
the ordinary tyre has a range of compounds and tread patterns which generally perform better when hotter, and by that I mean as they warm up, they have more grip. This is, for reasons of safety and consistent handling, a very linear relationship. So the answer is no - you won't find any tyres that "drop off a cliff" as the temperature drops.
remember that a tyre is only able to generate grip if it warms up. The slip in the tyre generates heat which allows the rubber to grip.
winter tyres are designed so that they have more grip at cooler temperatures - again a linear relationship between temperature and grip.
At the other end of the scale, grip does drop like a stone if the temperature goes above the level at which the tyre is happy - that's relatively easy to do.