One common theme is his relationship with his father. In many fanfictions, Todoroki's hurt is related to the emotional and physical abuse he suffered from his father. This can be explored in depth, showing how it affects his self - esteem and his ability to trust others.
You can start by looking on fanfiction platforms like Archive of Our Own (AO3). There are many talented writers who create Todoroki - hurt fanfictions. Just search for 'Todoroki hurt' in the search bar, and you'll get a list of relevant stories. Some might involve his physical pain from battles, while others could be about his emotional struggles, like dealing with his family issues within the storylines.
His friends would probably take care of him. Midoriya would be very eager to help, using his kind-hearted nature. He would make sure Todoroki is fed and comfortable.
One idea could be that all his friends at UA plan a surprise birthday party for him. They decorate the common room with his favorite colors, blue and red. Another plot could be Todoroki spending his birthday with his family, trying to reconcile with his father on this special day. Maybe he receives a heartfelt gift from his mother that has a deep emotional significance.
Todoroki's character development in time travel fanfictions can be quite profound. He may start as a somewhat aloof character, but through time travel, he begins to understand the importance of relationships more deeply. When he travels back and witnesses the relationships he could have had or lost, it makes him cherish the ones he has in his original timeline. His view of heroism also evolves. He realizes that being a hero isn't just about using his powers but also about making the right choices at the right time, which is especially highlighted when he has to make decisions that could change entire timelines.