Pokémon GO Battle League Down is unfortunately a recurring issue. There could be several reasons why it's down.
The Battle League has a history of instability The mode often has bugs or issues which could prompt the Battle League to be disabled temporarily as Niantic addresses the problem.
Pokémon GO Battle League Down: What's Going On?
When these happen, Niantic will announce that the Pokémon GO Battle League will be down for maintenance. The maintenance isn't scheduled, as the Battle League could go down for a few hours on consecutive days. Most of the issues Niantic Support work on relate to Battle League.
These issues are the most annoying, because it could go down whenever without much warning and mess with your competitive grind. There's not much you can do other than wait for Niantic to fix the server issues before you can battle in the League again.
Even when Niantic does an emergency fix, issues still arise after. Since the last fix, players have reported issues with lag and freezing.
Pokémon GO Battle League has never been stable, and unfortunately you just have to go in the mode expecting some issues. To keep up to date with server downtime, you can check the Known Issues page on the official support site.