Grease an 7 or 8-inch spring form pan with butter, and line the bottom with a circle of parchment paper. Set it aside.
In a large bowl add the cocoa powder and butter, and pour the hot coffee over them, whisking as the butter melts and the cocoa dissolves in the liquid.
Whisk in the brown sugar, milk and vanilla until the mixture is smooth and well combined.
In a smaller bowl, whisk together the flour, baking soda, baking powder, and salt.
Add the dry ingredients to the wet ingredients and combine them until the mixture is smooth and no streaks of flour remain.
Pour the batter into the prepared pan and cover with a double layer of aluminum foil, folding the sides of the foil tightly around the edge of the pan to form a seal.
Pour 3/4 cup of water into the bottom of the Instant Pot, place the cake pan on a trivet and lower it into the Instant Pot. Close the lid and turn the valve to sealing, setting the Instant Pot to cook on high pressure for 50 minutes.
Once the 50 minutes is up, let the Instant Pot release pressure naturally for 10 minutes, then release the remaining pressure manually and remove the pan from the Instant Pot. Remove the foil lid and let the cake cool in its pan on a wire cooling rack for about 10-15 minutes, and then remove it from the pan and place on the wire cooling rack until cooled completely.