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Arrange a rack in the middle of the oven and heat to 325°F.
Grease a 9-inch round cake pan or springform pan lightly with oil. Line the bottom with parchment paper and grease the paper if using a cake pan.
Whisk together the yogurt, olive oil, honey, thyme, and lemon zest in a large bowl. Add the eggs, one at a time, whisking well after each addition. Add the flour, baking powder, baking soda, and salt. Stir with a rubber spatula until the batter is almost smooth with just a few small lumps, but do not overmix.
Transfer the batter to the cake pan, and use a spatula to spread it out evenly. Bake until the top is lightly browned and a tester comes out clean, 40 to 45 minutes.
Transfer the cake to a cooling rack and let it cool for 10 minutes before removing it from the pan. Run a knife around the pan to loosen. If using a springform pan, unclasp the sides. Otherwise, flip the cake onto a plate and flip it back onto the rack or serving plate. Serve warm or at room temperature.