Preheat the oven to 350°F. Line the bottom of a 9-inch springform pan with parchment paper and butter the sides of the pan.
In a small bowl, combine the rosemary and zest and mash with 1 teaspoon powdered sugar. Set aside.
In a large bowl, whisk together 2 cups powdered sugar, flour, polenta, baking powder, cinnamon, allspice, and salt. Add the eggs and the rosemary mixture.
Using an electric mixer set on medium speed, beat until blended. Slowly pour in the oil, beating until smooth. Add the pumpkin and beat until smooth, about 2 minutes longer.
Spread the batter into the prepared cake pan. Sprinkle with the demerara sugar. Bake until a toothpick inserted into the center comes out clean, 40 to 45 minutes.
Transfer to a wire rack and let cool slightly, then remove the sides of the springform pan. Serve warm or at room temperature for up to 2 days, or wrap tightly and freeze for up to 1 month.
Notes
This recipe may also be made into muffins. Line 20 muffin cups with paper liners and bake for 18 to 20 minutes.