Heat oven to 325° F. In a food processor, pulse the gingersnaps until fine crumbs form (you should have about 1 cup). Add the butter and pulse to combine. Press the mixture into the bottom of a 9-inch springform pan (preferably nonstick), using a straight-sided dry measuring cup to help. Spray the sides of the pan lightly with the cooking spray. Place the pan on a rimmed baking sheet.
Using an electric mixer, beat the cream cheese and brown sugar on medium until smooth, 1 to 2 minutes. Beat in the eggs, one at a time. Add the flour, allspice, ginger, cinnamon, and salt and beat to combine. Add the pumpkin and beat to combine. Pour the mixture into the prepared pan and bake until set but still slightly wobbly in the center, 1¼ to 1¾ hours. (Tent loosely with foil if the top becomes too dark.)
Let cool completely in the pan, then refrigerate until firm, at least 2 hours. Run a knife around the edge of the cheesecake to loosen, then unmold.