When you are exercising your horse in a cold arena, you will need an exercise blanket until he is properly warmed up. Otherwise, you risk the chance of pulled and torn muscles because the body is unusually cold and is working too quickly. When cooling your horse from exercise, you can use one of your blankets to put on his hid quarters, preventing stiffening. I suggest wool or a fleece cooler, which actually wicks the sweat away and allows the horse to dry faster. Again, you have changed the horse's natural mechanisms for body heat, so you must be prepared to work with your horse.
Stable blankets have the same system.
If your horse is in a stall often, notice the temperature in the barn. Decide what materials are best for what your horse needs. Cotton is very cool, and will not warm a horse. There are good stable blankets available in light, medium, and heavy weights. You will need to watch how the temperature fluctuates in the barn to decide what is best for your blanketing system.