It's hard to answer any contract question without seeing it, but once the term of a lease runs out, generally the tenant becomes month to month and the tenant is free to leave (with 30day notice) and landlord is free to change any terms, including rent or security deposit upon reasonable notice by landlord.
The above answer is generalized reply to an question and is not intended to be legal advice or establish an attorney-client relationship with you. If necessary you should meet with an attorney and provide the attorney with all relevant documents and get an attorney opinion or advice on your situation.
Assuming you don't have a written agreement and you are renting on a month-to-month basis, your landlord can change the terms of your rental with 30 days notice, as Mr. Davidson pointed out. However, that 30 days cannot end before the last day of the next full rental term.
For example, suppose your rent is due on the first day of each month. On February 15 your landlord gives you notice that he is raising the rent. The new terms do not become effective until April 1. You have to be out of the apartment by the last day of the notice or you are considered to have accepted the new rental terms.
The answer to your question really depends on when they gave you notice of the change in rent. You should consult an attorney who can obtain more information from you and give you a more meaningful answer.
Once you become a "month-to-month" tenant, the landlord may change any of the terms by giving you 30 days notice. Pets can be a source of problems and expenses for landlords, and it appears the proposed changes reflect that fact. If you cannot negotiate something agreeable to you both, it may be time to start looking for a new place for you and your pet. Good luck!