Preheat the oven to 350°F (175°C). Grease an 8-inch round cake pan with coconut oil and lightly dust with gluten-free flour.
In a medium bowl, whisk together the gluten-free flour, baking powder, baking soda, salt, ground cinnamon, and ground nutmeg until well combined.
In a large bowl, beat the eggs. Add the coconut sugar, maple syrup, melted coconut oil, vanilla extract, and unsweetened applesauce. Mix until smooth.
Gradually add the dry ingredients to the wet ingredients, stirring gently until just combined.
Fold in the diced apples evenly through the batter.
Pour the batter into the prepared cake pan and spread it evenly.
Bake for 40 to 45 minutes, or until a toothpick inserted into the center comes out clean.
Remove the cake from the oven and let it cool in the pan for 10 minutes.
Transfer the cake to a wire rack to cool completely before serving.