Well, I think it's true that the more cold air that spills out of an
open refrigerator, the more warm air comes in to replace it, and you pay
to cool that warm air.
But, even though I haven't read every post in this thread, something I
haven't seen mentioned is the word "momentum" and any reference to the
driving force that would cause that air to gain momentum and spill out
of the fridge.
Air is actually a lot heavier and denser than most of us realize. It's
only because our bodies are also pressurized to the same pressure that
surrounds us that we don't feel the force of the air pressure acting on
us. In fact, for most of recorded history, people believe air didn't
weigh anything at all. In fact, it's both the mass of the air and the
tiny density difference between warm air and cold air that determines
the rate at which cold air would spill out of a fridge. The calculation
at the beginning of this thread presume ALL of the cold air spills out
to be replaced by warm air each time the fridge door is opened and I
don't believe that to be true. In order for air to spill out of an open
fridge, it has to move, and it takes time for that to happen given the
small difference in density which is the only thing causing the cold air
to gain momentum.
That is, cold air isn't going to pour out of a fridge like cold water
would. The small difference in density is going to result in the cold
air spilling out much more slowly than cold water would (for example).
I suspect the difference would be similar to what would happen if you
had a fridge full of cold water submerged in a warm swimming pool.
Opening the door would result in some cold water spilling out, but
nowhere near as much as would spill out if that fridge weren't submerged
in warm water. The important difference being the fact that the driving
force imparting momentum to the cold water is very much reduced.