2008 Chevrolet Cobalt P0411 Code: Cold Start Check Engine Light Issue After Air Pump Replacement

My 2008 Chevrolet Cobalt (52,000 miles) has been experiencing a recurring P0411 diagnostic trouble code. The check engine light activates within minutes of starting the vehicle during cold weather—specifically when temperatures are between 0°F and 15°F. This pattern has occurred four times since December, each time after a cold start. In every instance, the light comes on less than one minute into engine warm-up and typically goes out by the next day as the vehicle's ECU performs a routine secondary air system self-test, per the factory service manual. A local dealer diagnosed the issue and replaced the secondary air pump under warranty. However, after five days of cold weather in Ohio (with overnight temperatures around 10°F), the check engine light returned during a second cold start while idling. The technician noted that the original air pump was noisy and showed signs of binding during testing, which led to replacement. Despite this, the P0410 code persists under similar conditions. I'm wondering if there's a way to trigger a failure in the secondary air valve (or system) during cold starts—such as using a specific diagnostic procedure or engine warm-up protocol—to confirm whether a faulty valve is actually causing the P0411 code. Any insights from owners with similar issues would be greatly appreciated.