Your car is about 10 years newer than mine, so this might or might not be correct. As far as I know, the outside temp gauge is not related to the AC system.
I had a similar problem with my outside tempature gauge. I was able to fix it myself with advice from Barrie of this forum. In fact, I keep a file on my PC with some of the best advice from this forum for fast access.
This is a clip I cut and pasted from that file regarding the outside temp gauge.
"If your outside temperature gauge is not working the fault is most likely
with the sensor unit. You can test both the sensor and display by
connecting a 3,000 ohm resistor in place of the sensor. If the display is
working properly then your gauge (with power on) will give a readout of
approx. 77 degrees F (or 25C if you have a celsius gauge).
Having verified that the fault is with the sensor, you need only replace
the sensor thermistor (about $5). However, M-B sells only the complete
cable assembly (the sensor at the end of a long piece of wire, for
about $50), which requires about 2 hours to route - hence the $200
cost. To minimize this labor some mechanics will buy the assembly, cut
off the end, and splice it into the old cable. Spliced connections,
particularly in the engine compartment, are not a good idea! Instead,
you can easily replace the sensor yourself for $5 with no splices. Use a
negative temperature coefficient (NTC) thermistor of 3k at 25C, such
as Fenwal #192-302LET-A01 or Thermometrics #EC95F302W. They are
readily available from any electronics supply house.
If you require more detailed procedures on testing or installation, post
your questions and I'd be glad to oblige."
Now, I have had a hard time finding a thermistor in my area. I need to do mine again, so if you find one, plese let me know where you found it!
Thanks and good luck.
Kyle De Priest
2007 e320 Bluetec