When filling a constant volume gas thermometer, the gas should be...

A constant volume gas thermometer operates on the principle of the ideal gas law, $PV = nRT$. For the thermometer to provide accurate readings over a wide range of temperatures, the gas used must behave as closely to an ideal gas as possible.

Real gases approximate ideal gas behaviour at high temperatures and low pressures. Under these conditions, the intermolecular forces become negligible, and the volume of the gas molecules is insignificant compared to the total volume of the container.

Therefore, the gas should be maintained at high temperature and low pressure.