Preheat your oven to 350°F (175°C). Grease a 9-inch baking pan.
In a mixing bowl, cream together the butter and sugar until light and fluffy. Add the eggs one at a time, beating well after each addition. Stir in the vanilla extract.
In another bowl, whisk together the flour, baking powder, and baking soda. Gradually add the dry ingredients to the wet mixture, alternating with the sour cream. Mix until just combined.
Pour the batter into the prepared baking pan and spread it evenly.
For the crumble topping, combine the brown sugar, flour, cold butter, and cinnamon in a bowl. Use a pastry cutter or your fingers to mix until the mixture resembles coarse crumbs. Stir in the diced apples.
Sprinkle the crumble topping evenly over the cake batter.
Bake in the preheated oven for 45 minutes, or until a toothpick inserted into the center comes out clean. Let the cake cool in the pan before serving.