What is the name of the post-game area added in Pokémon Emerald?

The post-game area added in Pokémon Emerald is called the Battle Frontier.

Pokémon Emerald expanded the Battle Tower from Pokémon Ruby and Sapphire into a much larger challenge area. After defeating the Pokémon League Champion, players could access the Battle Frontier and compete in several specialized battle facilities.

The area offered rules and objectives different from the main story. Winning battles earned Battle Points, which could be exchanged for prizes useful both inside and outside the facility. This gave experienced players a substantial reason to continue playing after completing the Hoenn adventure.

The Battle Frontier is sometimes confused with the Battle Tower, because the tower is one of its facilities. It is also distinct from later locations with similar names, such as the Battle Resort in Pokémon Omega Ruby and Alpha Sapphire.