1. In a large pot, combine the sugars, cornstarch, cinnamon, nutmeg, allspice, and water. Cook over medium heat, stirring constantly, until the mixture thickens and begins to bubble.
2. Add the lemon juice and sliced apples to the pot. Stir to coat the apples with the sauce.
3. Bring the mixture to a boil, then reduce the heat and simmer for 10 minutes, or until the apples are tender.
4. Ladle the hot apple pie filling into sterilized canning jars, leaving ½ inch of headspace. Wipe the rims of the jars clean, then place the lids on top and screw on the bands until fingertip tight.
5. Process the jars in a boiling water bath for 20 minutes. Remove the jars and let them cool completely before storing.