In terms of keeping it cool, that probably depends on how good your freezer's insulation is. If it's really good, it probably doesn't take too much energy to keep things cool (air or otherwise). If it's not so good, you'll incur a constant "your freezer sucks" tax.
In the case being discussed, we're talking about freezing stuff every day, and letting it thaw in/near your bed during the night. So the cost would definitely be there.