Yes, K&N intakes can actually boost power and torque; in fact, it's guaranteed. Installing a cold air intake is one of the quickest ways to increase airflow, power, and throttle response. The K&N air filter promises to deliver more horsepower, and it did. The Crosstrek produced 164.3 hp and 142.5 lb-ft of torque. The gains may be small and you may not feel the extra power.
More air means more power, and oversized K&N high-flow filters will allow for greater airflow. Replacing the OEM air filter with a K&N high-flow filter will add power and increase acceleration, but the biggest gains will come from installing a complete cold air intake system. You'll get improved throttle response, slightly better mpg, and maybe a few horsepower if you use a pull-down panel filter. You'll get a little more from a cone filter with a & thermal protection tube. A cold air intake, if available, will make the most of it.
Of course, as you climb the ladder of admission types, the price increases accordingly. Research has found that the paper used in filters is an important part of the problem - the paper is excellent for filtration but bad for airflow. Volkswagens change their filters every service interval, while Hyundai and Maruti change their filters every alternative service interval. It is essential that the air drawn into the engine is clean and sufficient to allow complete combustion. Cotton allows much greater airflow than paper, but it risks allowing contaminants to enter - but K&N has it covered with their secret sauce: filter oil.
The powder contains silica, which are small particles of hard rock - if allowed, I would gladly sandblast the intake system while your engine draws in air. On the other hand, a K&N air filter is said to be composed of layers of cotton gauze placed between sheets of aluminum wire mesh. An air cleaner has a conflicting requirement that is not to prevent air flow to the engine (if technically possible, allow an “unrestricted” flow of clean air to the engine). In addition, if the filter becomes clogged with particles for a period of time, less air (and therefore oxygen) is supplied to the engine. Personally, I use K&N air filters because I live in a dusty area and I usually spend more on paper filters than on oil filters. For me, K&N filters don't actually filter particles enough for me to take a chance, so I simply put on a new OE filter every 12 months or so. Standard air boxes are designed to have no dead spots so that turbulent air simply swirls and does not enter the engine.
Therefore, running without the filter allows dirt to contaminate and damage the sensor, which can lead to incorrect airflow readings - which, as you know, is vital to fuel ratio calculations. Most standard airboxes draw air from the front or wheel well of the car, where the air it's much colder than in the engine compartment. It suggests that the increase in power could be due to less filtering - especially in the case of the cheaper CarQuest filter - although because it had no way of testing the filtering process, that aspect was not observed.