It's quite possible that the coolant temperature sender (note, not coolant temperature sensor as that's a different kettle of fish ) is knackered. No need to start the engine - test it cold, but if you remove the plug from the sender (usually a single wire) earth it, the gauge should go straight up, or straight down depending on how it works, certainly when its disconnected the guage should read 'zero'.

Does the needle go up with the ignition on or only when the engine's running?