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Unfortunatly renting something isn't the same thing as leasing to own. Otherwise I would own like 4 houses by now.
With that said, you can purchase DVR's from Tivo and use it with Cox. They also have different purchase options, but most require a monthly subscription. Also, if your looking for encrypted channels, you would need a cable card, and possible a tuner adapter.
You could also buy a PVR. A PVR is like a DVR, but no TV tuner inside. It requires you to plug something into it for a video source. You could rent a HD cable box, connect it to the PVR, and record that way. That would save you the monthly DVR fee from Cox, and the hassle of needing a cablecard or tuner adapter. It would mean two boxes though, and possible confusion switching between the two.The main reseller of PVR's is Hauppauge.
Last, if your really looking to geek out, you can basically build your own DVR/PVR. You would need some kind of TV tuner PCI card (or external adapter) and special software that runs on your computer. You would need to dedicate a computer to that function though, and this is NOT for the ameatuer. If you haven't build a computer from scratch, then I don't suggest this option. Otherwise, see here for more info.
There is a fairly inexpensive alternative if all you need is an HD Set-top-box tuner. There are no associated monthly Guide fees.
Samsung GX-SM530CF Cable Box and Streaming Media Player with Built-In Wi-Fi www.amazon.com/Samsung-GX-SM530CF-Streaming-Player-Built-In/dp/B00EYO241Q/
You would need to rent a CableCARD from Cox for $2/mo and you get the Tuning Adapter for free.http://ww2.cox.com/residential/support/tv/article.cox?articleId=b8889a60-d706-11df-66a2-000000000000