IME, frig thermostats fail O/C, so the frig never runs at all, but this one has failed C/S, so it runs all the time. Thinking about it, could this be because the operating fluid/gas has leaked out of the 'stat so that it never opens the contacts that control the compressor?
I've ordered another 'stat, anyway, but I just wondered ...
(Interesting price range for replacements, from £6 from someone on eBay up to > £100 on some web sites...)