could it be the thermostat?

No not unless the motor only runs for a very short time then stops. Thermostat turns the motor on when cabinet is too warm. Might be a build up of ice stopping cold air from circulating. Try switching off at wall & leave doors open for 12 hours (overnight) to melt away any ice. Best done when fridge & freezer are almost empty. If that doesn't work then it may of lost the gas needed to cool the fridge. You will need a service company in to regass it.

Lincoln Appliance NZ
October 2009