One possible reason could be his feeling of being an outcast. In the original stories, Loki is often treated differently in Asgard due to his nature. He might have felt like he didn't truly belong, which could fuel his hatred. Another reason could be envy. Maybe he saw others in Asgard having things he desired, like power or recognition, and his envy turned into hatred.
He could meet the Frost Giants. Since he has some connection to them already, they might welcome him or at least be intrigued by his presence. He could learn more about his heritage from them and maybe even form an alliance.
One reason could be that Loki feels constantly overshadowed by Thor. In the Asgardian hierarchy, Thor is often seen as the golden child. Loki, being the more complex and misunderstood one, might run away to find a place where he can be his own person without living in Thor's shadow.
One possible plot could be Loki leaving Asgard after feeling constantly overshadowed by Thor. He might leave in search of a place where his skills and intelligence are truly valued, perhaps finding a hidden realm where he can start anew and build his own following without the judgment of the Asgardian court.
Maybe Loki was too obvious in his admiration. He could have been constantly quoting Odin's teachings or trying to imitate him in public.
In the fanfiction, Loki's hatred for Asgard might make him more of an anti - hero. In the original canon, he has complex relationships with Asgard, but not always pure hatred. Here, his actions are likely to be more extreme as he is driven by this strong negative emotion.
One common theme is redemption. Loki is often seen as a character who has made mistakes, and in these fanfictions, Asgard's protection of him is part of his journey towards redemption. For example, Thor's unwavering support helps Loki to face his past actions and strive to be better.
It could potentially make his relationships more complex. Some might view him with suspicion, thinking he has ulterior motives for being an Odin fan. Others might try to use this knowledge to their advantage, perhaps to manipulate Loki or gain favor with Odin.
One reason could be that they see Loki as a victim of circumstances. Maybe they've come to understand that his actions were often influenced by forces beyond his control, like Thanos' manipulations. So they decide to defend him in Asgard.
Loki could use his magic. His pregnancy might give his magic a different kind of energy. He could create illusions to show the builders what the finished parts of new Asgard could look like, helping them to work more efficiently.
One great 'Asgard loves Loki' fanfic could be 'The Redemption of Loki in Asgard's Eyes'. It shows how the people of Asgard gradually come to understand and love Loki despite his past misdeeds. Another could be 'Asgard's Embrace of Loki' which focuses on the emotional journey of Loki as he feels the love from his home realm.