Some encoding happens during video processing on the capture hardware itself, since a USB2 device like the HD PVR can't fit all of the uncompressed video data over that type of connection. OBS would still encode it with x264 prior to sending it out, so you would see no performance benefit there.
Even if you used a capture card that supported hardware encoding like the Live Gamer HD, and even if OBS supported that card's hardware encoder, and even if you could both capture with the card and use its hardware encoder at the same time (you can't without using AVerMedia's own software), it would still be pretty inferior to x264 - somewhere between the ultrafast and superfast presets. Capture cards are great for getting console footage, and great when you're using a 2PC setup where a second PC is encoding all of the video from the first PC. They are generally a waste of money for single PC streaming unless you have some very specific capturing needs.