Your Landlord can always ask for additional deposits, but that doesn’t mean they are entitled to it.  So, this means that by asking for an additional pet deposit, you landlord is not necessarily breaking a law; however, they are probably not able to demand it either.
If your landlord agreed to the additional pet and then asked for a deposit much later, I would tell them to go back to the original agreement in which they had already authorized the new dog without additional deposits.
Don’t be afraid to tell you apartment manager that you can’t come up with the money.  They may drop it, or offer you something that is more fair.  You can always offer to pay them in smaller installments or a lesser amount.  Its probably better to cooperate than it is to fight.   Even if they are dis-organized and not entitled to the pet deposit, you will still be wise to work with them.

Yep! You should work with them on this. The worse thing to do would be to cause your landlord or property manger to have a reason to not renew your contract. The simple fact is that you are in violation of the lease, so be very cautious about how you move forward. Don’t get hostile! Find a amicable solution that works for you and your apartment manager.