I imagine it is like putting a thermostat in your house that controls both yours and your neighbor's house. You wouldn't be able to tell the difference in your house (refrigerator portion) but your neighbor (the freezer portion) might tell a different story. The fridge portion will be controlled as you desire, but the freezer will come on and off on the schedule determined by the fridge temperature.
It may keep a constant temperature. It could also very likely have huge fluctuations in temperature due to differences in insulation, surface area, volume, etc.
I'd say get yourself a thermometer, preferably one that records high and low temperatures and monitor it. I would be very interested in knowing myself what happens.
Or, if you know anything about appliance repair or electronics, you may be able to hook the external thermostat to just the refrigerator portion and keep the freezer as-is. Or get 2 external thermostats to make 2 different aging areas with different conditions for different cheese types. Personally, I am not electrically or mechanically inclined, so I don't know how to do it, how difficult it would be, or if it is even possible.