First off... whatever works best for any one GM's game is always the right answer. What follows is the official in-Golarion answer, but feel free to adjust as you wish for your game.On Golarion, paladins don't HAVE to worship a deity, but the vast majority do. Those who do worship a deity need to worship one that's either lawful or good... and it's best if it's a deity who's lawful good. Golarion has paladins of all the lawful neutral, lawful good, and neutral good deities (although in some cases, like Irori, those paladins are relatively small in overall number compared to deities like Iomedae, Abadar, and Sarenrae, who have a LOT of paladins).
There are no paladins of Gorum, because a paladin simply can't worship Gorum and remain lawful good—and being lawful good is what being a paladin is all about. Becasue...
If you're worshiping Gorum like a religious character should, and are correctly following Gorum's teachings... you're not interested in good or evil, and you're doing a lot of actions in your religious rites and practices that are fundamentally chaotic in nature, and as a result, your alignment will shift to chaotic relatively swiftly. And once your alignment shifts away from lawful, you become an ex-paladin.
If you're keeping your lawful good alignment by doing things that are lawful and good but still claim to be worshiping Gorum, you're really only doing lip-service to Gorum and not really worshiping him at all. At BEST you're a heretic, but with that huge a difference in alignment, it's more likely you're blaspheming. In either case, claiming to belong to a group (in this case, the faith of Gorum) but behaving fundamentally unlike a member of that group is expected to behave is a chaotic act, and as such you'll soon shift to a chaotic alignment anyway. And as soon as you shift away from lawful, you become an ex-paladin (and are likely to be hunted by Gorum worshipers eager to put you down for being a blasphemer).
You CAN be an antipaladin of Gorum, but that's an entirely different character!
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