There could be. Try searching on different fanfic websites. Some might be hidden gems. It really depends on how popular the Adam x Lute pairing is among the Hazbin Hotel fanfic community. If it's not a very common pairing, it might take more effort to find really good ones.
Common themes may include the exploration of their personalities. Since they are characters from Hazbin Hotel, their unique traits are likely to be delved into. Also, themes of loyalty and trust could be present, as their relationship develops. They might face challenges together that test their bond, which is also a frequently seen theme.
Charlie is often paired with Lute. Since she is the owner of the Hazbin Hotel and very passionate about redemption, she might see Lute's music as a tool for helping the sinners, and they could work together on different projects.
A plot could also revolve around Lute using his music to try and rehabilitate some of the more difficult sinners in the hotel. His music has a special power that can soothe or inspire change, and he tries to use it to make the hotel a better place for everyone.
Charlie would be a great character to pair with Adam. Charlie is all about redemption and Adam could either be a foil to her or be influenced by her positive attitude. Their interactions could be really interesting as they have different outlooks on the whole idea of sin and redemption in the Hazbin Hotel world.
One idea could be Adam's backstory exploration. Maybe he had a different life before coming to the Hazbin Hotel universe and that could be delved into. For example, his family, his upbringing, and how those things might have shaped his current self.
Redemption is also a frequent theme. Given the setting of Hazbin Hotel, the idea of Adam finding redemption through his relationship with the reader is quite popular. It could involve him learning about love and acceptance in a new way.
In the 'Hazbin Hotel' fanfiction, the relationship between Lucifer and Adam could develop in many ways. Adam could be initially afraid of Lucifer, but as he spends more time in Hell and gets to know Lucifer better, he realizes that Lucifer has his own complex emotions. Lucifer, on the other hand, might be used to being feared but Adam's different reaction, like showing pity or trying to understand him, could make him more receptive to Adam. They could also be brought together by a common enemy or a shared goal, like trying to improve the conditions in a certain part of Hell. As they work towards this goal, their relationship would grow from suspicion to trust and perhaps something more than just friendship over time.
Another option is to check out Fanfiction.net. Sometimes, there are some great Hazbin Hotel Adam x Reader fanfictions hidden among the many works on the site. It might take a bit more searching compared to AO3, but it's still a good place to look.
One idea could be that Adam and Eve end up in Hell and have to adapt to the chaotic and sinful environment of Hazbin Hotel. They could meet the main characters and get involved in their schemes to rehabilitate sinners.