Bradypodidae is the family consisting of the three-toed sloths, the one we most associate the word "sloth" with. As far as we know, they exist only in the tropical forests of Latin America and never climbed upwards to the more temperate United States.
The only sloths instead to cross the border into the United States were the few families and subfamilies of ground sloths (giants, Harlans, Shastas and Jeffersons). Being ground sloths, they did not share the three-toes' adaptations for living and instead roamed great distances for a wider variety of plant foods.
But let's say that there is a species of three-toed sloth that did make the crossing and is now munching leaves in the forests of the Appalachian Mountains. Would its lifestyle still be the same as that of its tropical cousins, or are the personalities of temperate deciduous forests so different that it would need to give its slow, arboreal, folivorous ways and be a ground sloth itself?