Hmmm, I suppose it's possible it could be an exhaust issue; it did pass NV air inspection in Jan of this year, so I don't think it's a gutted cat. I took it in for air inspection here in CO but apparently the traction control won't turn off even though the dash says it's off when you toggle it. For that reason they've not been able to test it yet. I'm pretty sure I know the culprit in that problem, just need to fix it so they can try again.
I have to say, it's a very pronounced knock sound when you hear it in person. When you say it might be an exhaust leak I think you're getting the impression from the recording that it's more of a resonance or hollow sound, but it's not.
All that said, unless it indeed has something to do with the exhaust warming up, it does seem odd that it goes away once hot. I'll be putting in 0w-30 full synthetic tomorrow and will update.