The sensors resistance to earth drops as it gets hotter (its a thermistor). If it says you are hot when you apply power it kinda suggests that the resistance is very low/zero.
Either the wire is touching earth somewhere (the block, where it passes through the bulkhead etc) or the sensor has died.
To check, simply remove the wire from the sensor and the needle shouldn't move. if you touch the wire to ground (ie the engine etc) then the temp gauge should spring into life and show HOT
You haven't said what engine/car you have and there are a few versions of temp sensors, look up AMR1425 or AMR3321 oe ERR2081. thjey are all around the £5 delivered.
If the sensor is still saying HOT when its dissconnected, then you have a short to ground on the wire somewhere - or - the guauge is forked (rare).
If you disconnect the sensor wire from the guage you should be able to suss out the issue.