Kudos to Cosmic Charlie. From another thread, same story:
"The system will display the last known outside temperature when starting the vehicle and may need to be driven several minutes before the updated temperature is displayed. Engine temperature can also affect the
displayed temperature; therefore, temperature readings are not updated when the vehicle is not moving."
That's all the Owner's Manual says about it. But, I found this explanation on another forum:
Here is the ambient temperature sensor behavior for the JK:
Vehicle with the key off for more than 4 hours...The CCN will update the ambient temperature reading immediately.
Vehicle with key off for more than 2 hours but less than 4 hours....The CCN will update the ambient temperature reading immediately after vehicle reaches above 20 mph (with no duration constraint) and will update every 500 ms thereafter.
Vehicle with the key off for less than 2 hours...The CCN will
display the last temperature reading (which is stored in memory) just prior to the ignition being turned off. The CCN will not update the value in the display until the car has been driven at 20 MPH or above for 3 consecutive minutes. If the vehicles speed and drive time does meet the criteria the CCN will continue to display the stored value until the criteria is met. Once the criteria is met, then the CCN will update the display every 500 ms based off of the ambient temperature sensor output.