If the motor is running it seems unlikely the problem is the thermostat.
My first thing with a new freezer is to plug it into my energy monitor so I have data to refer back to, but the motor with most fridge freezers runs for around 1/3 of the time, if it does not stop, then there is clearly a problem. I have had it where the thermostat failed, and motor never stopped, and found got to around -26ºC, my problem at that time was how to measure temperature inside the freezer, so I heated some water up with salt, so it was as saturated as possible and put it in the freezer, it froze solid, so clearly below freezing point of brine which is -18ºC.
If the motor is turning on/off then could be thermostat, or the defrost heater cutting in too often, the freezer will likely draw around 100 watt cooling or less, but the defrost heater is more like 160 watt.
A frost free freezer works by having the cooling parts behind a panel, and a fan moves the cold air from behind the panel into the freezer when running, but fan does not run when defrost heaters cut in, so if the fan fails, then freezer will not cool. But the thermostat may be triggered giving one a false thought about thermostat failing.
Often I found fan stopped by ice build up, so cure was empty freezer, prop door open and aim fan at freezer to melt any ice, not just the ice you can see, but ice behind the divider, so leave fan running for an hour after last ice has melted. I found this did work with one fridge freezer, however latter found reason for the ice build up was the insulation had failed on the unit, ice was forming between cracks in the insulation causing the heat to enter freezer through the ice, once melted freezer would work OK for 6 months until ice built up again. As a freezer it was scrap, but worked very good for brewing beer in.
