Which ideal gas law does a constant volume thermometer work on?
Gay-Lussac's Pressure Temperature Law.

The law simply states


  The pressure of a gas of fixed mass and fixed volume, is directly proportional to the gas's absolute temperature.


As telling temperature rises, the gas in the thermometer has its pressure increased in accordance with the law. Measuring this pressure gives us the temperature as per the ideal gas equation, or by using Gay-Lussac's law directly, taking the pressure ratio against a set base temperature and pressure.