By virtue of his own nature being what it is, the cold itself does not harm Aslan or hamper him from doing what he usually does. He possesses a warmth and inner light all his own that dispels the cold and keeps him as he has always been.

Others are not so fortunate.

Pod will find him making his way slowly through a thicket, breathing softly on trees, plants, and other living things as he passes by. Even in the bitter cold, something of spring follows in his steps, and ice slides quietly from branches, birds, and other beasts in his wake.

Magic is what is causing this and so it is magic that must set things right.