Rather straightforward ..
You can carry a copy of your most recent pay stub and other supporting information for your own comfort - in all the travels I did on AP, they never looked at any of the stuff. But read on for a couple of scares ..
Scare 1
Once, while coming back once, the guy at Delhi (India) airport wanted to make a copy of my AP .. and the carrier did not have a copier at that terminal. I hesitated a lot - he was going to walk out to another terminal and get back in 10/15 minutes - if he did that is, and I would be left with no legal re-entry document. What if this was a racket? What if he never came back? (Luckily, and I did not disclose this to him - I had another AP inside my handbag). With my hesitation - he said, sir, you are looking at the possibility of having to stay back in case you don't give me your papers to copy. He was with the airlines (Continental) and seemed to be known around the place, so I let him take my papers (did not have a choice, did I?). But he did come back .. no racket, no conspiracy.
Scare 2
The dumbo at the immigration on the way back (again, same trip - Delhi Airport) asked me about my visa. I said - I have AP. He looked at the AP and read it may be twice. Then he called some one over and asked "Wasn't there a recent circular about not allowing folks with AP to board flights to the US?" - while I listened. Oh man! What is this? The other officer said "I have not seen any such thing, AP is OK". Relief? No .. this idiot persisted "No, I have seen the circular that the AP is no longer valid". The other officer said "It is fine, as long as the date allows you re-entry". With one further study (oh yes, may be he had an exam coming up on it the next day .. can you believe it?) of the AP, he very reluctantly let go uttering something like "You should be thankful, because I have tried to help"/ Yeah, right - moron! All this while, I stood there quietly - no point in annoying any official who can make sure that you miss your flight!