I just don’t buy this twice as efficient bullshit.

Do you understand how heat pumps work? The heat you're drawing on is the the heat of the outside compartment on the outside, therefore the heat moved to the inside can be more than just the heat equivalent of the electric energy you put in. That's how these achieve more than 100% efficiency, in general. In fact if the outside isn't so cold outside they can achieve 300%-500%.

Now the trick to moving heat from a cold outside compartment to a warmer inside compartment lies in the compression. If you draw even a moderate amount of heat energy into your medium, then compress it, it will turn quite hot allowing you to dump heat into your warm inside compartment. Then as the medium flows out you let it expand and it turns really cold, cold enough that it can draw in heat from the cold outside. But the lower the difference in temperature of the outside air to your expanded medium gets the less heat you can transport per unit of time, that's why we're only looking at 200% here.