Temp gauge is not working

What make , model and year vehicle ! Sounds like the instrument cluster is bad or no data from serial data network ! No you know what that is ? C.A.N is on 2003 and newer but older vehicles had other networks . How to Diagnose Controller Area Network CAN

The PCM on a 2004 Ford Mustang is a combined control unit, consisting of the engine control unit (also known as an ECU) and the transmission control unit.------------------------------------------------------------------------------------------
Launch X431 Master

I would suspect the outise air temp sensor is defective, replace it. It is located in front of the AC condensor core.

Coolant Temperature Sensor I would take the vehicle back to where it was serviced and try to ask them about what may have gone wrong to cause this problem.
The Intake Air Temperature sensor should also be checked.

The following is the procedure for servicing the coolant temperature sensor: Removal & Installation 3.8L Engine To Remove:

1. Drain the coolant.
2. Disconnect the engine coolant temperature (ECT) sensor connector. ECT sensor 3.8L
   
   
3. Remove the ECT sensor.

To Install:

1. Apply thread sealer P/N 12346004 or equivalent to the engine coolant temperature (ECT) sensor threads.
2. Install the ECT sensor. Tighten 18 ft-lb (25 Nm).
3. Connect the ECT sensor harness connector.
4. Fill the cooling system.

5.3L Engine To Remove:

1. Drain the coolant below the level of the engine coolant temperature (ECT) sensor.
2. Disconnect the ECT sensor electrical connector. ECT sensor 5.3L
   
   
3. Remove the ECT sensor from the cylinder head.

To Install:

1. Clean the threads in the sensor mounting hole.
2. Apply sealer P/N 13246004 or equivalent to the threads of the sensor.
3. Install the sensor and tighten 15 ft-lb (20 Nm).
4. Connect the ECT sensor electrical connector.
5. Fill the cooling system and check for leaks.

best guess is that you lost ground to the cluster, start at the battery and dissassemble the grounds and clean them, nest step is to locate the grounds under the dash, good luck

COULD BE THE FUEL SENDER OR YOU HAVE A POOR GROUND POINT IN THE CABIN.
WEIRD AND MULIPLE MALFUNCTIONS ARE USALLY A POOR GROUND.

if you have a digital dash it could be going bad,you can repalce it with a new on or go to the dealer and they can send it out and have it repaired the other cause can be either a loose connection for your dash or a bad ground.check your battery connections as well as other grounds at the fron of the radiator support there is a bolt with little ground wires remove the bolt clean with sand paper and replace.With out being there this is a little hard to diagnose,good luck.PS has this car ever been hit in the front?

For general awareness, the outside air temperature sensor is located above the inside mirror.
Try to reset the temperature measurement unit: 1. Switch engine on. 2. Press and hold clock minute (min) and hour (h) buttons simultaneoutly longer than 3 seconds. The display will show new temperature measurement unit. 

hi, all the gauge are place in cluster,in front of your steering wheel,easy for your visual,if you have your manual you can determine if your gauge is not original or factory install,ty

Page 106 from the owner's manual. It says (if equipped), it's in the MESSAGE CENTER. Press the INFO button (you should see 3 buttons INFO, SETUP, RESET, pressing INFO will toggle the odometer display between a number of things like odometer, distance to empty, outside temperature, etc.

Good luck, if you still need help don't hesitate to stop by your local dealer and have them show you.