Well, in such a fanfiction, he might control space through his magic wand. Maybe he uses a special spell that he discovers or invents, which allows him to manipulate the fabric of space, like creating portals or warping distances.
He might get involved in a battle across different galaxies to protect the wizarding world from an intergalactic threat. His power to control space would be crucial in quickly transporting himself and his allies to different battlefronts, and also in creating strategic barriers in space to fend off the enemy.
Often, Voldemort remains the main antagonist. Even if Harry can control space and time, Voldemort's thirst for power and his army of Death Eaters are still major threats. He might try to find ways to counter Harry's new abilities or use them for his own evil purposes. For example, he could try to capture Harry and force him to use his space - time abilities to open portals to places where more dark magic items are hidden.
One interesting scenario could be Harry using his space - control ability to instantly transport himself and his friends to hidden, safe places during the Death Eater attacks. For example, he could open a spatial portal to a secret forest glade where they can set up a temporary base. Another could be using time control to go back and prevent certain tragedies, like Sirius's death. He might be able to freeze time right before Bellatrix casts the fatal spell and save Sirius.
Maybe he accidentally finds out when he is near some magnetic objects in Hogwarts. For example, he might notice that he can make the metal keys move towards him without using magic spells he already knew. It could be during a time when he is exploring an old, forgotten part of the castle where there are a lot of metal objects and strange magnetic fields.
In some fanfictions, it could be through an accidental encounter. Maybe he was in a situation where metal objects started to react to him unexpectedly, like when he was near a bunch of keys and they suddenly started to move towards him. This made him realize that he had this strange new power.
Perhaps he has a strange dream where he's in a world made entirely of shadows and he can manipulate them freely. When he wakes up, he starts to experiment and finds out that he can actually do the same in the real world. He might start with small things like making the shadow of a quill move on his desk, and gradually discovers the full extent of his power.
Well, it might develop as a result of some new magic he encounters. There could be an ancient spell or a magical artifact that gives him the power to control sand. He then has to figure out how to use it. He may start with simple things like making sand swirl around his hand, and as he gets more used to it, he can do more complex things like creating sand barriers or even using the sand to communicate in some way.
Perhaps it's through an accidental encounter with a powerful magical plant. He tries to interact with it in a normal way but ends up influencing its growth and behavior. This unexpected event makes him aware of his unique power to control nature. For example, he might touch a withering plant and suddenly it springs back to life and starts to grow rapidly under his touch.
In the 'Harry Potter Space Empire Fanfiction', for the characters to adapt, they first need to accept the vastness of the space empire. Harry, having faced Voldemort, would draw on his experience of dealing with great evil when encountering new, powerful adversaries in space. Hermione would start researching about the different magical cultures across the empire, learning new spells and incantations unique to each planet. Ron would be the one to bridge the gap between the more studious characters and the more adventurous ones. He'd be out there making connections, while also relying on his friends' knowledge. They'd all have to re - think their understanding of magic, as the space environment might require new ways of casting spells, perhaps using starlight or the energy of black holes as magical sources.