Greetings!

The symptoms could be a cable or cold solder joint. You said that it works fine on AC power. The fact that it is not going off after the key is pulled may or may not mean anything - some cars will keep the accessories "live" for a period of time after the key is pulled.

Have you noticed any other symptoms in the car? Battery icon? Is it only when it is really hot / cold out? You may want to open the unit and verify the cable connections, and to see if it is from the batch of bad cables.