Trivia: After Dr. Young is killed in the warden's office, go back down the stairs but do not go through the archway. Instead, after going down the steps, turn right and you will come to a blank wall. Spray the explosive gel on the wall three times and then detonate it. The wall will break apart and you will find a secret room. Go inside and you will find blueprints for the Arkham City Prison Complex.
Trivia: One of the collectable resources in the game are the Riddler trophies, and one of them is located near the graveyard outside of the asylum. After collecting the trophy, look closer and you'll find a grave belongs to Amadeus Arkham. Amadeus Arkham was the founder of Arkham Asylum.
Trivia: Scarecrow threatening to dump his latest fear toxin into Arkham's water is exactly what he planned to do in both Batman Begins and Batman: The Animated Series episode "Dreams In Darkness."
Answer: It's not the actual Batcave, it's a secondary location Batman set up in case of emergency. Oracle actually makes a comment along those lines the first time you access it.