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Gather all ingredients. Preheat the oven to 350 degrees F (175 degrees C). Grease a 9-inch round cake pan with butter and line the bottom with parchment paper.
Peel (or leave skin on) and core apples; cut into cubes between ¼ and ⅜-inch in size. A little smaller is okay, but if your cubes are any larger, they won’t get tender enough when baked. Add to a large bowl and toss with apple cider vinegar.
Mix together flour, baking powder, and salt in a small bowl.
In another bowl, beat butter, ½ cup white sugar, and brown sugar with an electric mixer until light and fluffy. Beat in 1 egg on high speed until it is fully incorporated and the mixture is smooth. Add the second egg and repeat.
Mix in rum, vanilla extract, and ½ of the flour mixture. When the flour disappears add the rest, and fold in with a spatula.
Fold in apples until evenly mixed into the batter. Transfer batter to the prepared pan, making sure all the batter is pressed in firmly. Smooth top and dust with remaining 3 tablespoons sugar.
Bake in the preheated oven until a skewer inserted into the center comes out clean, about 45 minutes.
Remove cake from oven and let cool in the pan for about 30 minutes. Invert cooled cake onto a wire rack, place a serving plate on top, and flip back over. Dust top with powdered sugar and serve.