An operating deepfreeze, like any other electrical appliance or light, brings energy into the house that, first, does its work, then dissipates as heat, as energy can be neither created nor destroyed, by the law of conservation of energy.

The deepfreeze mechanism works like an air conditioner, except it pumps heat from inside a box into the room in which it's located, as opposed to pumping it from inside the house to the outside world. I'm more familiar with the numbers for an air conditioner, so I'll use those, but the idea is the same. A 6,000 BTU/hr air conditioner typically might require 1,000 Watts of electricity to run. That 1,000 Watts, converted to Imperial units, is about 3400 BTU/hr and has to go some place. What happens is that while 6000 BTU/hr is extracted from the room, 9400 BTU/hr is sent to the outdoors.

If the deepfreeze door is opened, the built-up cooled air inside mixes with the warmer air outside the box (but still inside the house), to make the living space a bit cooler. But if the deepfreeze is left on, the mechanism will merely continue moving heat around in a continuous cycle while adding to that heat by the BTU/hr equivalent to its Wattage at about 3.4 BTU/hr for every Watt it uses.  Only that one-time shot of cold air is "free". It wasn't really free, but the room has already received the heat generated by cooling that air, but at least you can get some of that already-generated coolness.