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.
A theme of acceptance is also prevalent. Loki is often seen as an outcast, and Odin's love can represent acceptance of Loki's true self, whether it's his magical abilities, his different nature compared to the other Asgardians, or his past mistakes. This acceptance can be a slow and arduous process in the fics, full of emotional ups and downs.
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.
One great 'odin loves loki fanfic' is 'A Father's Unspoken Love' on FanFiction.net. It delves deep into Odin's complex feelings for Loki, exploring their relationship from a more emotional and psychological perspective.
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.
Loki might react with anger. He could seek revenge against Odin, plotting ways to undermine his power or expose his wrongdoings. This reaction is common as it shows Loki's pride and his unwillingness to be a victim.
One common theme is redemption. Loki often tries to redeem himself in Odin's eyes. Another theme is family. Their relationship as father - son is explored, whether it's a strained one or a reconciled one. And power is also a theme. Loki's desire for power and how Odin's power affects their relationship.