While the butter cools, begin peeling and coring the apples.
Pour the sugar into the mixing bowl and stir into the butter to combine.
Add the rum, vanilla, and eggs. Stir to combine.
Fold in the flour, baking powder, and salt.
Slice the apples very thin and add to the batter. Gently fold to coat the apples.
Pour the batter into the springform pan and bake for one hour. Check for doneness by inserting a toothpick into the center. If it comes out clean, the cake is done.
Allow to cool for at least one hour before serving.