Preheat the oven to 350 degrees F (175 degrees C). Butter a 9-inch springform pan.

Combine sugar, eggs, and vanilla extract in a bowl; beat with an electric mixer until smooth. Mix in rum. Mix in 1/2 of the butter.

Combine flour, baking powder, and salt together in a separate bowl. Add 1/2 of the flour mixture to the egg mixture; mix well. Mix in remaining butter and flour. Fold in apples.

Scrape batter into the prepared pan. Smooth the top gently.

Bake in the preheated oven until top is golden brown and a toothpick inserted into the center comes out clean, 50 to 60 minutes.