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This is a tough situation to address. As long as you have notified apt. management about your dog and paid the pet deposit, you should be ok. However, all tenants should expect peace and quiet in their own apartments. It is very well possible that your neighbor has greatly exaggerated the issue.
I would advise talking to the neighbor and the landlord to see if it was a one-time issue or if there are other alternatives. Can you move the dog's cage to a different room where either it doesn't hear as much outside noise and therefore barks less, or moving it further away from the offended neighbor's place? Maybe if you leave a tv or radio on for the dog at a reasonable volume, the sound will drown out any outside noise that would cause the dog to bark (like people talking or car doors slamming)...
It seems unfair for the landlord to threaten eviction for a one-time occurrence. See if there is any possibility to smooth things over. (I usually prefer the peaceful solution that doesn't require legal action.) If that doesn't work, perhaps you can check the lease to see if this eviction threat is consistent with the terms of your lease. If it is, you may not have much recourse. If it is not, consult with a local attorney.