While I am not licensed to practice law in your State, it appears to me that what we are dealing with here is the issue of "damages" incurred by the landlord as a result of your failure to timely vacate the premises in accordance with the Lease Agreement. The first thing I would do is go back and read the Lease Agreement itself. There may be a provision which sets forth the specific rights and obligations of the parties in the event you fail to timely vacate. If the agreement states that the lease automatically renews for 30 days you could be on the hook. If there is no such language, then you need to look into the attendant circumstances. If the Landlord had agreed to re-let the premises to new tenants who could not take possession and had to make other arrangements until the 1st of September, you could be responsible for the entire month's rent. However, if the Lease is silent on this issue, and there was not another tenant waiting to take possession, I think you are in a better position to argue that you should be responsible for 1/31 (August is 31 days) of the month's rent.