Preheat the oven to 350°F (175°C). Grease and flour an 8-inch square baking pan.
In a medium bowl, whisk together the all-purpose flour, baking powder, baking soda, ground cinnamon, and salt.
In a large bowl, combine the granulated sugar, brown sugar, and melted unsalted butter. Mix well until smooth.
Add the eggs one at a time to the sugar and butter mixture, beating well after each addition.
Stir in the vanilla extract.
Alternately add the dry flour mixture and the sour cream to the wet ingredients, beginning and ending with the dry ingredients. Mix just until combined.
Fold in the diced apples gently until evenly distributed.
Pour the batter into the prepared baking pan and spread evenly.
Bake in the preheated oven for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove from the oven and allow to cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before slicing into 8 pieces.