Yes, you can freeze apple sponge cake. Let the cake cool completely before cutting it into pieces and then wrapping those pieces in plastic wrap. Place those pieces in the freezer. You could also freeze the whole pan of cake if you have room in your freezer. Just be sure to let the cake AND the pan cool completely before covering it with plastic wrap and setting it in the freezer. We do not recommend freezing apple cake with frosting. Make the frosting right before serving after the cake has thawed out.