What's this thing in a peltier cooler?

It's a thermostat, originally known by the Klixon (now Sensata) name. They are based on a bimetallic disk, which is why they are cylindrical.

Some types have a reset switch (a button in the center of the hidden side in your photo) and do not automatically reset, but most are simple thermostats with a fairly wide (maybe 5 to 15 degrees C) hysteresis. Below is a photo of a resettable type with mounting ears from this website:

They are rather common in consumer appliances- cheap, simple and rugged.

I think Bimetallic Thermostat is the proper term. These use two metals with different expansion, so the bands made from these bend at certain temperatures. I guess this part is cylindrical because the band is rolled up inside. But it's hard to tell from this picture.