Well, in the context of the 'Lost Hero' world, there's no canonical evidence to suggest that the gods read fanfiction. Fanfiction is a creation by fans for fans, and it exists outside the official narrative of the story. So, probably not.
Jason could be reading The Lost Hero fanfiction to see how other fans view the characters. Different writers in the fanfiction community may have very different takes on the heroes and villains in the story, and he might be interested in seeing those diverse perspectives.
You can try websites like FanFiction.net or Archive of Our Own (AO3). These platforms have a large collection of fanfictions, including those for 'The Lost Hero'. Just search for 'The Lost Hero' on their search bars and you'll find a variety of stories.
Maybe he just loves the original The Lost Hero so much that he wants more of it. Fanfictions can be a great way to get extra content when you're really into a story.
Some mortals read The Lost Hero fanfiction because they are huge fans of the original book. Fanfiction allows them to explore different storylines and character developments that may not be in the official book. It gives them a chance to delve deeper into the world created in The Lost Hero.
Leo could read the fanfiction to gain more understanding of his own character. Fanfictions often explore different aspects of a character's personality. By reading about himself in different scenarios within the fan - made stories, he can learn things about himself that he may not have realized before. For instance, he might discover how others perceive his relationships with his friends or how his actions in the original story are interpreted in different ways. This can help him grow as a character.
The Olympians' reactions to reading The Lost Hero fanfiction would vary greatly. Hermes would be the first to spread the news among the gods. He loves gossip and new things. Demeter might be indifferent as she is mainly concerned with the harvest. But for Artemis, if the fanfiction in any way misrepresents her values or her hunters, she would be furious. Hephaestus might be interested in the technical aspects of how the fanfiction was written, like the structure and the use of language.
If the gods read the 'Lost Hero' fanfiction, they might be amused. They could find some interpretations of their characters quite creative or, on the other hand, completely wrong. Maybe they'd be angry if the fanfiction depicted them in a negative or disrespectful way.