Sounds like an exhaust leak to me, either where the exhaust manifold meets the cylinder head or further back. If you take off the metal shield (it's 3 10mm bolts if I recall correctly), you can get a better look and (very carefully) feel with your hand around the manifold with the car running. You should feel the pressure of the exhaust escaping when you get near where the leak is.

 

That is strange sounding almost like a miss. From my history with fits I would check to make sure spark plugs are tight (doesn’t sound like that problem but maybe) then take the serpentine belt off (only run it a short time like this as the water pump won’t be running) and see if the problem is still there.

A small section of hose on your ear can help give you directional hearing or a screwdriver with a long tip resting on the suspected component.

 

Second for loose spark plug as exhaust leak is more consistent and tend to go away at normal temperature. If plugs are OK, take the drive belt off and run engine for 10 seconds to rule out belt driven accessories.