It could be due to a spell. There are many dark spells in the wizarding world that might cause a person to feel cold. For example, some curses can drain the warmth from a body. If Harry was targeted by such a spell, he would be cold. Also, it might be related to his emotional state. If he was feeling lonely or scared, it could make him feel cold inside, just like when people are in a bad mood in real life, they might feel colder than usual.
I don't think so. Harry Potter is known for his courage, loyalty and compassion in the official story. While fanfiction can explore different interpretations, making him cold - hearted goes against the essence of the character that most fans are familiar with. So it's not likely to be a common theme in fanfiction.
One idea could be a crossover where Captain Cold somehow ends up in the wizarding world. Maybe he's on the run from the Flash and uses a magical artifact in Central City that zaps him into Hogwarts. He could initially be seen as a threat but then team up with Harry Potter to fight a common enemy, like a dark wizard who has acquired some sort of technology -mixing magic and science in a dangerous way.
It could also be about a new character in a fan - created story within the Potter world. This character might possess a power related to cold fury, like being able to freeze things when enraged. And the fanfiction could explore how this power affects the character's relationships with others in Hogwarts or the wizarding world at large.
One example could be a fanfiction where Harry has been through so much trauma that he becomes cold and emotionless. In this type of story, he might isolate himself from his friends at Hogwarts. For instance, he stops joining them for meals in the Great Hall and spends most of his time alone in the Room of Requirement, poring over dark magic books not for evil purposes but to understand the nature of the darkness that has affected him. Another example could be a story set in an alternate universe where Voldemort won, and Harry, as a result, has his spirit broken and becomes cold and unresponsive to the world around him. He might be forced to serve Voldemort in some capacity, but his coldness makes it hard for Voldemort to truly break him completely.
In some fanfictions, a cold Ravenclaw can be portrayed as having a sharp tongue. They use their intellect to make cutting remarks when others don't meet their standards of intelligence or logical thinking. This coldness could be a defense mechanism, protecting themselves from getting too involved with others. For example, when a Gryffindor challenges their view on a magical concept, they reply with a scathing rebuttal.
Since it's a fanfiction, it could have the main characters from the original series like Harry, Hermione, and Ron. They might be dealing with situations that involve cold fury in ways that J.K. Rowling didn't explore in the original books.