Am I obligated to pay a full day's rent for an additional five hours in a furnished apartment, and is that considered a debt?

You are required to pay rent for the extra period you remained in the apartment after the lease agreement expired. If the contract or common practice dictates paying a full night's rent for late handover, then you must pay it. If it is impossible to reach the landlord to pay the due rent, you may give it as charity on his behalf.