The only reason the box has a lid on it is so you can change the filter
and the air coming into the filter is from the outside of the engine compartment. Cooler air will make the engine more emission friendly, but also restricts the airflow, so when you take the lid off you get hot air from the engine compartment but you also get more unrestricted air. I would check out K&N filters they have a freeflowing filter set up with a reusable filter element. A little pricey but they guarantee more horsepower.
http://www.knfilters.com/FIPK/fipk.htm