Audiobooks, unlike your other iTunes purchases, cannot be re-downloaded from the store. On a Mac, you can dig into your iTunes folder and move the DRM-protected files to another authorized machine. iOS devices, however, lack a file system that the user can access, so if you never sync with a Mac, you could run into an issue like I did when I had to swap out my iPhone 5.
A couple of weeks ago my iPhone 5's power button started acting strange. It appeared to be a mechanical issue, so I went to the Genius Bar at my local store and was told they've seen this happen before (it was likely shearing stress from removing a protective case).
Of course, they don't repair iPhones in the store. Instead, they give customers a new piece of hardware then use a backup to bring it back to where you left off with your old one. Since I used iCloud, PhotoStream and my iTunes purchases are available in the cloud, I figured I'd be fine. The only problem was that I had purchased a $24 audiobook via iTunes on my iPhone -- and that doesn't get backed up in any way. Worse, since the rules state you can only download the audiobook once, I would have had to re-purchase the book to download it again. I had bought the book about three days earlier and only listened to a small portion.
This was completely unacceptable, so I emailed Apple support. At first, they failed to understand what I was talking about. They thought my download had failed somehow, and suggested I download in iTunes and re-sync, and they would re-enable the download. That seemed fair enough, but the presumption that I had a personal computer to sync with struck me as a bit odd. Didn't Apple say we live in a post-PC world? Also, I don't sync my iPhone with anything. Like a fool, I thought iCloud backup and iTunes in the cloud had me covered. Clearly I was wrong.
Once I explained that I didn't sync, the support staff told me they would make it available for download on my iPhone -- but the downloads screen never showed the audiobook. In the end, I wound up buying a nano to sync with, and then Apple refunded my money for the audiobook. An expensive lesson, but a lesson learned. Apple is still addicted to iTunes as a digital hub for all your media.
While this was happening I tweeted some progress and a number of people relayed similar issues and frustrations. Apple support seems ready and willing to try to make the downloads available again, although I wish it was automatic, like everything else I encountered during my restoration process (well, not logins -- boy that's a fun way to spend your afternoon). Unfortunately, the rules about audiobooks aren't set by Apple. Still, if this happens to you, reach out to Apple support immediately. Or, better yet, buy your audiobooks somewhere else. This "post-PC" era has yet to begin.