It's popular because it adds a new layer of drama. Fans love to see their favorite character, Percy, in unexpected and difficult situations. Betrayal is a very strong emotion, and when combined with a crossover, it allows for more complex storylines.
One good example could be a crossover where Percy is betrayed by a character from another series, like Harry Potter. Maybe in this fanfiction, Percy is tricked into believing a false prophecy by a wizard who wants to use his powers for their own gain. The story could explore how Percy copes with this betrayal and tries to find his way back to his true self.
Percy might feel a deep sense of hurt. He values friendship highly, and this betrayal would make him question who he can trust. He could become more guarded and less likely to trust new people easily. Reyna would be more focused on the bigger picture. She would see the betrayal as a threat not just to herself but to the stability of the demigod world. So she would start gathering her allies, those who still believe in her, and form a strategy to counter the false accusations and the forces behind the betrayal. She would be determined to prove that she and Percy are not the traitors they are made out to be.
One aspect could be that in many Percy Jackson - related fanfictions, Percy being betrayed by the gods often leads to a complete shift in his character arc. He might become more cynical or distrustful. For example, he could start to question his previous heroic deeds and wonder if he was just a pawn in the gods' games all along.
At first, Percy would likely be angry. He's a fighter, and being betrayed by someone he loves would fuel his rage. He might go after Annabeth to demand an explanation. But as the reality sinks in, he could become very sad and withdrawn. He might isolate himself from his other friends as well, feeling like he can't trust anyone anymore. He would constantly be thinking about what he did wrong to make Annabeth turn on him.
He could be angry. Percy might lash out at those who betrayed him, feeling a deep sense of injustice. He would probably be determined to prove his innocence and get revenge on those who wronged him.
Apollo might be the betrayed god. He could have been tricked by another god into betraying Percy. For example, Apollo might have been promised more power or a higher status among the gods if he turned against Percy. But then he realizes too late that he's been used and that he's actually betrayed someone who could have been an ally.
One idea could be a crossover with Star Wars Rebels. Ahsoka could be betrayed by someone she thought was an ally in the fight against the Empire. Maybe a new character who was secretly working for Thrawn all along. This would lead to a complex plot where Ahsoka has to not only survive but also figure out who she can trust again.
In some fanfictions, Annabeth might betray Percy for power. Maybe she gets lured by some dark force offering her great power and in order to get it, she has to turn on Percy. It could also be due to a misunderstanding. For example, she might be tricked into believing Percy has done something really bad and as a result, she betrays him without realizing the truth.