Not all stovepipes register the same temperature when used with identical stoves and fires. This partly is because different stovepipes are made using different metal alloys, thicknesses and coatings, resulting in slight differences in pipe temperatures between different pipe brands. Despite these slight differences, the outside of a fireplace's double-wall stovepipe should fall between certain temperature to avoid problems. You can purchase thermometers and attach them to the outside of your stovepipe in order to judge the outer temperature effectively. You can place a thermometer on the outer wall at any point where it's accessible, though placing one inside the inner wall might be best done by securing a high-temperature thermometer in a clothes hanger and hooking it inside the shaft of the stovepipe if it's accessible.
If a double-walled stovepipe regularly falls below 230 degrees Fahrenheit during a fire, this is a sign of poor burner performance. It means that the fire is not burning efficiently and not producing enough heat. Instead, it is probably producing other things instead: dangerous soot and creosote that can clog your chimney. Lack of airflow and wet fuel, among other things, can cause these lower temperatures.
If a stovepipe reaches temperatures higher than 475 degrees Fahrenheit, this may be a sign that the fireplace is burning too hot. Exhaust particles may not be a problem, but overheating carries dangers of its own. It may set creosote deposits on fire, damage ignition systems or simply waste energy. The heat escaping through your stovepipe cannot warm your house and may be costing you extra money in heating bills.
You can measure stovepipe temperature by using a stove pipe thermometer to monitor conditions. You can attach these thermometers to the surface of the pipe or in some cases mount them with a safety screw or drilled hole. Always let your fireplace run for at least an hour before checking temperatures. Initial temperatures can fluctuate widely compared with the steady temperatures the fireplace will reach later on.