What would make a Volvo 240 DL 1986 temperature gauge read hot even without the engine running?

Check the temperature sensor. You should be able to find a resistance value in your manual. If it is within specs, check to make sure the wire to the sensor is not grounded. Quick check would be to unplug the wire from the sensor. If the temperature goes down to cold it is probably the sensor. If not, there could be a short in the wire to the sensor. Worst case scenario is that the guage itself is faulty but the sensor or the wire is more likely.