Final answer:
According to the lease agreement, the landlord does not have to refund any rent to Bob.
Step-by-step explanation:
According to the information provided in the lease agreement, if Bob gives a 30-day written notice to vacate the apartment on June 30th and pays rent for the entire month of June, the landlord does not have to refund any rent to Bob.
The lease agreement states that the premises shall be considered vacated only after all areas are clear of Bob's belongings and keys are returned to the owner. Since a new tenant moves in on June 15th, Bob is still responsible for the rent for the entire month of June.