Sift together the baking powder, baking soda, salt, flour, five-spice mixture, and ginger. In a medium bowl, mix the tea and applesauce. In a mixer fitted with a whisk attachment, whip the eggs and sugar until very light and fluffy. With the mixer running at medium speed, drizzle in the oil and mix. Add 1/3 of the dry ingredients and 1/3 of the tea-applesauce mixture and mix. Repeat twice more, using up all the ingredients. The batter will be somewhat thin. Pour the mixture into the prepared pan and bake until firm to the touch and split on the top, and a toothpick inserted into the center comes out clean, 70 to 90 minutes. Set the pan on a wire rack and let cool 30 minutes, then turn the cake out onto the rack, peel off the paper, and let cool.