Combine sugar, pumpkin pie spice and corn starch in a large bowl.
Peel and core apples; cut 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 pie crust. 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 10 minutes. Reduce heat to 350ºF and bake an additional 40 to 50 minutes, until golden brown. Serve warm with vanilla ice cream, if desired.