Preheat oven to 350°. Grease an 8-inch square, metal baking pan. Add flour, sugar, baking soda, pumpkin pie spice and salt to pan; mix.
Combine applesauce, oil and molasses in a measuring cup. Stir into flour mixture with a rubber spatula until blended; smooth top. Bake until springy, for about 25 minutes. Let cool completely, for about 30 minutes.