Mix sugar, pumpkin pie spice and corn starch in a large bowl. Cut apples into 1/4-inch thick slices and place in a separate bowl. Sprinkle apples with lemon juice and rum; add to dry ingredients. Stir to combine. Place in piecrust. Dot with 2 tablespoons of butter. Drizzle with caramel sauce.
Stir flour and brown sugar together in small bowl. Cut in 1/4 cup butter until mixture resembles coarse meal. Sprinkle evenly over top of pie.
Bake in a preheated 400 degrees F oven for 10 minutes. Reduce heat to 350 degrees F and bake an additional 40 to 50 minutes, until golden brown. Delicious when served warm with a scoop of vanilla ice cream.