Unless you're in extreme conditions, cooler temperatures have very little effect on most tyres for the road, winter tyres will be a softer compound to provide more grip not to generate heat in them to soften them further to generate grip. Road tyres mainly suffer because they get too hot when you're working them hard not because they're too cold, rallying on snow and ice it is surprisingly easy to overheat tyres. Cooler temperatures aren't an issue unless you're on track day/race tyres, cut slicks are used on the monte carlo rally and that's held in January, well up in the alps, much of it on icy roads.