The only June measurable snow of record at nearby Big Delta, Alaska, was nearly a century ago in 1917. But those seem to be very, very bad readings -- way above freezing, said Alaska-based climatologist Brian Brettschneider.
Your lease is a legal contract that commits you to pay rent for the right to live in the landlord's apartment. Most leases and most jurisdictions require tenants to give notice to their landlords before they vacate the property. On the plus side, the obligation is mutual, which means that your landlord cannot order you to leave without notice.
Unless you agree to move out, the new owner cannot evict you without bringing an eviction proceeding in Housing Court. There is no pre se rule as to how long you have to stay. However, as a month-to-month tenant, 30 days is ordinarily sufficient for giving notice. Yes, you are obligated to pay rent for the time that you are an occupant of the premises. A security deposit is just that, security; it is not rent.