Preheat oven to 350 F and take your crust out of the freezer to slightly thaw.
In a large pan melt the 4 tablespoons butter on medium heat then add the apple slices. Allow the apples to cook and turn slightly brown for 7 to 10 minutes. You don’t need to cook them until they turn crispy just until they start to get a slightly golden tint on them. Sprinkle in the sugar, brown sugar, cinnamon and rum and stir for 5 minutes. The mixture will thicken slightly.
In a small bowl or cup add the cornstarch to the water and whisk with a fork so the cornstarch is completely dissolved. Now pour it into the apple mixture. Begin to stir the mixture for another 5 minutes and the apple filling should be nice and thick. Once the mixture has thickened remove from heat and set aside.
In a medium bowl, add the ingredients for the topping (flour, brown sugar, sugar, pecans, and butter). Use a fork to blend all of these ingredients together. It should look like large clumps of sand.
Pour the apple filling into the pie crust then sprinkle the topping on top of the apple filling. Place the pie on a baking sheet. Place inside the oven and bake for 45 minutes.
Remove from oven and allow to cool for about 20 minutes and that will also allow the filling to set a bit more. Then you are ready to serve.