Preheat the oven to 350°F, and grease a 9x13 baking dish or pan with butter or shortening. Set aside.
Stir the cocoa powder and hot water together in a medium bowl until the cocoa has dissolved and the mixture is smooth. Set aside to cool slightly.
Whisk together the flour, baking powder, baking soda, and salt in a medium bowl, and set aside.
Combine the granulated sugar, oil, and vanilla extract in a large bowl, and mix with an electric hand mixer until fluffy. Mix in the peanut butter and cocoa and water mixture until smooth, then add the eggs and mix until well combined.
Gradually add the dry ingredients to the wet ingredients, and mix until the batter is smooth and no streaks of flour remain. The batter will be thick and wet. Pour the batter into the prepared baking pan, and spread it into an even layer.
Bake for 25-30 minutes, or until a toothpick inserted into the center comes out clean and the cake bounces back quickly when lightly pressed in the middle. Remove from the oven, and let the cake cool to room temperature in the pan before frosting.