FWIW, according to my AP Stylebook, the rule is capitalize when used as a formal title before a name (Pope Paul came, saw, went....); lowercase in all other uses (Afterwards the pope did x, x, z...)
But what does AP know :)
"Pope" is part of the man's name. It signifies that he is the direct successor to the apostle Peter, the first pope. The AP might be clueless, but it is more likely that they are aware of this and choose to disregard it.