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When you say that the dog was "not in your care" - was the dog still your property?
If this was before you owned the dog, you would not be liable, but if you owned the dog and it was merely being watched and cared for by another individual, you would remain liable as the dog's owner.
You would have a cause of action against that caretaker for indemnity, but the landlord has a claim against you as the dog's owner.
You are responsible for your dog. (Even though it was being cared for by someone else).
The person caring for the dog is liable to you for reimbursement (the legal term is "indemnity") - you can charge them for the bill that you have to pay to your landlord - but your landlord has a right to charge you.