2002 WRX - engine sounds like a diesel one

A coupe of thoughts here: It could be an exhaust leak. These often quiet down considerably after the engine gets hot. It's easy enough to put the car on ramps and crawl underneath and tighten the nuts which hold the header pipes to the heads. If an exhaust leak has a ticking sound, it's usually an individual exhaust port that's leaking. It could also be a cracked pipe. It could also be a valve that's somehow gotten loose. I think these are the removeable shim type adjusters, though not sure. Is this still under warranty??

An additional OT comment: I read about a study a long time ago, I think it was done by Ford, that indicated that ring wear was about

800% greater during cold engine operation than during hot operation. This was due to 1) The extra fuel being delivered during cold running would wash the oil off of the cylinder walls. 2) The cylinder walls, when hot would cause any fuel landing on them to instantly flash into vapor. When cold, condition #1 would be made that much worse. In the days fo carburators, most chokes would open even though the engine wasn't warming up properly and the engine would run crappy and/or stall out. But nowadays the coolant temp sensor will cause the "correct" amount of fuel to be injected for any given engine temp. I had a cousin who used to start his car and go back inside to have a last cup of coffee. The car sat and idled for 5 - 10 minutes 5 days a week (this was in mild San Diego). At 60k miles his rings were shot and the Nissan was burning a lot of oil. My personal strategy is to let oil pressure build (10 seconds) then put it into gear and gently drive until I see the temp guage start to move. Even under a light load the engine will warm up much faster.

There was a group of the early 2002's that had questionable timing belt tensioners, your local service department may know more.