It sounds like the thermostat could be inaccurate if that's the case. I can't picture the exact thermostat housing. Turn the power to the cabinet off, if the light bulb cover is attached to the thermostat housing remove it and see if there is a screw hidden in there, some can be a fiddle to remove.
Sounds like you could have a faulty thermostat. Is the control knob within the fridge compartment? I take it from what you say in your post the compressor is not running all the time?
What brand of paint would you recommend to paint our soffits?
Intended to use Dulux Weathershield (Green Glade) but after reading the reviews have had second thoughts. Its a country house at around 120 years old.