Okay, so between posting this and now, I did some testing on my equipment.
My first (and only) test was to do with my cheap PS3 Component AV lead, and so I played a bit with it plugged directly into my TV. In about 40 minutes worth of play, I had the screen blackout for a second or two on three occasions. It wasn't too frequent, probably not 7 minutes, but they weren't far off each other. I'd say about 5-10 minutes in for the first occurance, about 10 minutes after was the second, and the last one was 10-15 minutes after that.
So my guess is that the signal was being lost, which is the first black flash you see in the video. The Hauppauge HD-PVR realizes this a second later, which is when the screen goes black the second time as it displays "No Signal." which is followed instantaneously by another flash of black, when the Hauppauge realizes the signal is back and readjusts itself to the resolution again.
I have ordered the official Sony Component AV Cable from play.com and will update this thread with my findings shortly after it arrives (it's currently being picked and packed, so if it is dispatched today, I expect to receive it no earlier than Thursday) and will try it out for a few hours before updating this thread. I am hoping it is the lead, because it doesn't paint a very good picture of the Hauppauge HD-PVR to pull it out of the box, brand new, and have this happen to it.
Should this not work, I will send it back to Amazon and ask for a replacement.