G'day Al, one type of machine (spectrometer) I worked on had a +- 0.1 degrees C oven controller that was based on a mercury thermometer with two wires going through the glass. There were several different thermometers available from the list on the machines circuit diagram. They listed ones rated at 32, 34, 36, 38 or 42 degrees, so I'd expect there to be a much wider range available, but I don't know the original vendor.
These thermometers would get a bubble forming once every few years, but cooling them or heating them to the point where the mercury sat either in the bulb or the expansion chamber on the end. Heating them with a cigarette lighter as was often done to clear them was a trick to not overheat them.
Chemical Laboratory suppliers could stock them or another (more expensive) model that has an adjustable contact with a temperature scale read on the side and a screw to adjust the contact wire length. I picked up mine in a laboratory closedown for nothing, but without the external power controller. When I use it to control the slab temperature in my wood stove hot water and heated floor system, I'll make a drive circuit to operate the pump relay. It's in a 1 inch glass tube about 12 inches long.
You could also use a cheap digital inside outside thermometer with alarm and modify it to drive a relay. They can often be found in variety stores.
If using one to control a heater element, add an overtemp cut-out to protect against unattended fire.
You also want to mount your sensor in a place that doesn't get the draft from the air conditioner. Do not have the air conditioner switching on and off to quickly as repeated quick starts can cause compressor damage. Aim for a couple of minutes between starts if you can when operating on tight temperature specs. This will cause more temperature change, but will allow the unit to last a season or two. Some air con controllers include a time delay.
Hope this helps, Peter