You cannot charge extra rent or an added security deposit for service animals (trained) or emotional support animals (not trained). Only animals defined as 'pets' may result in higher rent or security deposits. The animal is treated as a necessary medical device and you are required to make that reasonable accommodation.
Unfortunately, the tenant and their medical provider get to make this decision about what is a pet vs. a medical device, not you.
Does the practice get abused a lot? Oh yes. Can you do anything about it? Not really. However, if the animal causes damage to the property, if they fail to clean up after it, or if it attacks your or neighbors you would have the same remedy as you would with any other animal.
The other thing that may help is if your insurance provider specifically states they will significantly increase your insurance premium or cancel your policy due to that specific animal (not just the breed...THAT animal), you may be able to characterize that as an unreasonable accommodation. So for example if your insurance carrier sends over an inspector who see an aggressive pit bull ESA lunging and barking at every passer-by, they may send you a letter saying you must remove the animal or lose your policy. You could use that as evidence in your favor.
It's still murky water at this point legally speaking. Not many court cases have been fought yet defining all of the parameters. My advice is tread carefully and err on the side of allowing folks to abuse the system, as nutty as that sounds. You do not want to be the person who gets to pay the bill so the rest of us get legal precedent set.
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