When a tenant, being lawfully notified by his landlord, shall fail or refuse to quit the demised premises and deliver up the same as required by the notice, or when a tenant shall give notice of his intention to quit the premises at a time specified, and shall not deliver up the premises at the time appointed, he shall, in either case, thenceforward pay to the landlord double the rent which he should otherwise have paid, to be levied, sued for, and recovered as the single rent before the giving of notice could be; and double rent shall continue to be paid during all the time the tenant shall so continue in possession.