Preheat oven to 325 degrees F (165 degrees C). Butter a 10 inch tube pan and coat with white sugar.
In a large bowl, cream together the butter, applesauce and sugar until light and fluffy. Add the eggs one at a time, beating well with each addition (about 4 minutes), then stir in the vanilla, vanilla paste, rum and butter extracts. Reserve 1/4 cup of flour for later, and sift together the remaining flour, baking powder, cinnamon and salt. Gradually stir into the creamed mixture. Use the reserved flour to coat the chopped peaches, then fold the floured peaches into the batter. Spread evenly into the prepared pan.
Bake for 70 minutes in the preheated oven, or until a toothpick inserted into the cake comes out clean. Allow cake to cool in the pan for 10 minutes, before inverting onto a wire rack to cool completely.