Preheat your oven to 350 degree Fahrenheit (175 degrees Celsius). Butter and flour a 9-inch springform cake pan and line the bottom with parchment paper for easy removal.
Peel the apples and slice them thinly with a mandoline or a sharp knife, removing the cores. Add them to a small bowl and set aside.
In a large bowl, cream the butter and sugar until the grains of sugar start to disappear. Add the eggs (one at a time), vanilla extract, and the rum extract (or rum).
Add the dry ingredients on top of the butter, sugar, and egg mixture. Whisk the dry ingredients together a little bit before you stir everything together well.
Fold in the apple slices.
Add the batter to the prepared pan and spread it out evenly on top.
Bake at 350 degree Fahrenheit (175 degrees Celsius) for about 35 minutes, until the cake is tender and a toothpick inserted in the middle of the cake comes out almost clean.
Allow the cake to cool in the pan for about 10-15 minutes before transferring it to a wire rack to cool completely. This helps the cake set and prevents it from breaking.
Slice and serve when the cake is cool.