I'm well aware that the default answer to this textbook default question is "it doesn't work", but still, I believe it does.
To cool the insides of the fridge, the compressor must do work, and since the efficiency isn't 100% you are constantly warming the whole room to cool it's insides, the winning move here is simply turning the fridge off. However, let's suppose the fridge must stay on, wouldn't it be better to open the door?
In other words: Isn't opened fridge turned on better than closed fridge turned on for the whole room temperature?
Best Answer
No, you are making the fridge do extra work, so more energy is coming in (through the plug) as the pump continues to run since it's not reaching it's cold point. A normal operating fridge does not manifest cold air; it just pumps all the heat out of the inside of the fridge.
The action of pumping the heat out also has heat as a byproduct (which is waste heat from the power coming through the plug as it does work with the pump). So you're really just generating more heat and moving heat around.