I don't know what the air flow rate is with the cold air intake vs. the OEM intake, but I would have to imagine it's a pretty significant difference. especially if you have exhaust, headers, etc. to push even more air flow through the motor.
The stock fuel injectors are tuned and set up for the stock intake. If you're doubling the amount of airflow, it may be just borderline between where it kicks a code or does not kick a code. The stock ECU/Injectors is engineered and designed to work. There is a certain amount of headroom for safety, but once you start doing a certain amount of upgrades, it can bypass what the factory curves are able to support. Cold air is more dense, so would be more air to the fuel mix (leaner). This doesn't mean you're still not running lean when it's warmer. The ECU has a threshold of when it kicks a code. IT may just be that when it's warm, it's inside this threshold (barely) but the colder denser air is enough to put it over the threshold and kick the light. The only way to compensate would be to go with a little larger injector and/or fuel management system.
But first, I would make sure your injectors are clean, fuel pressure is per spec, etc. Go through and do a full tune up, make sure the MAF/MAS is clean, etc. I definitely would not recommend running it hard or much with it being lean or bad things can happen.