Preheat your oven to 350°F (175°C). Grease and flour a 9-inch round cake pan.
In a mixing bowl, whisk together the flour, sugar, cocoa powder, baking powder, baking soda, and salt until well combined.
In another bowl, beat the softened butter until creamy. Add the eggs, cooled espresso, and vanilla extract, and mix until fully incorporated.
Gradually add the dry ingredients to the wet ingredients, mixing until just combined. Do not overmix.
Pour the batter into the prepared cake pan, smoothing the top with a rubber spatula.
Bake in the preheated oven for 30 minutes, or until a toothpick inserted in the center comes out clean.
Remove the cake from the oven and allow it to cool in the pan for 10 minutes, then transfer it to a wire rack to cool completely.