In most states, the landlord has the right to keep the deposit when the tenant breaches the lease. The tenant is responsible for all the rent till the lease mature, theoretically. But, usually as long as they pay for the last month, you let them go.
Be very careful of double rent. For example, if your previous tenant moved out at the 10th and paid for the whole month. Then you find someone else to move in on the 16th and pay half month's rent. You have to pay back the previous tenant that half month's rent. Otherwise, you are in trouble if one of them sues you.
