Heat the oven to 350°F. Grease an 8½ by 4½-inch loaf pan with cooking spray or softened butter. Line the bottom with a long sheet of parchment paper that extends up and over the long sides of the pan (like a cradle for your cake).
Whisk the oil, sugars, and vanilla for 30 seconds in a large bowl. Whisk in the eggs, one at a time, and then the apple sauce. Sprinkle the baking powder, cinnamon, cardamom, nutmeg, and salt into the bowl, one at a time, vigorously whisking after each. Gently fold in the flour. Scrape the batter into the prepared pan, smooth the top, and lightly sprinkle with Turbinado sugar. Bake for about 60 minutes, rotating at the halfway point. The cake is done when a wooden skewer inserted in the center comes out with a moist crumb or two.
Let cool for about 20 minutes, until you can safely lift the cake out of the pan by the parchment overhang without burning yourself. Run a butter knife around the edges if it resists. Let cool to room temperature before slicing and serving. Keep the cake on the counter wrapped for up to 3 days.