For some reason this car really doesn't like warm starts or something. Just like you say, if I shut my car off for a short period of time then start it, it is LOUD and the exhaust is wide open regardless of drive mode. It chills out after a few minutes or if I start driving it. I haven't noticed any thing that sounds like a misfire during that though.
The other thing you might be talking about: the car will give some good sounding pops and bangs if you actually rev it in neutral or park, but that's just because it's hitting the rev limiter. This is a nice sound but if someone wasn't aware it could sound like something is broken.
The other similar sound is what "RJD" is saying where the car will start with the exhaust open, then very shortly after it will close, but it should not happen multiple times during the same start. Your clip isn't working for me so I can't verify if that's what the sound is but you will hear that sound on both cold and warm starts.