First thing you have to understand is that a termination of tenancy and an eviction are two different things. And, yes, your month to month tenancy can be terminated for any reason or no reason.If I have done nothing wrong, as a tenant, do I need to leave? Do I have any rights? Or is the notice for vacating by Sept. 30th all he needs to do?
The AZ landlord-tenant statute provides:
Paragraph B applies to your month to month agreement and the LL has certainly complied by giving you almost 7 weeks written notice.33-1375. Periodic tenancy; hold-over remedies
A. The landlord or the tenant may terminate a week-to-week tenancy by a written notice given to the other at least ten days prior to the termination date specified in the notice.
B. The landlord or the tenant may terminate a month-to-month tenancy by a written notice given to the other at least thirty days prior to the periodic rental date specified in the notice.
C. If the tenant remains in possession without the landlord's consent after expiration of the term of the rental agreement or its termination, the landlord may bring an action for possession and if the tenant's holdover is willful and not in good faith the landlord, in addition, may recover an amount equal to not more than two months' periodic rent or twice the actual damages sustained by the landlord, whichever is greater. If the landlord consents in writing to the tenant's continued occupancy, section 33-1314, subsection D applies.
Paragraph C is the eviction that can occur if you remain in the place past September 30. You should avoid eviction at all costs because eviction goes on to your credit reports, trashes your credit score, and makes it more difficult to find another rental if future landlords check your credit.
As for the AC issue, my own AC went out several years ago in August when it was 120 degrees outside. I was 4 days without it until my AC guy could get a replacement ordered, delivered and installed. I don't see 6 days as being so egregious that you would win anything in court, since the landlord had good reason for the delay and provided you with a portable unit.
That he is terminating your tenancy over this is rather stupid but it is his right and he's likely to pay for his mistake when his next tenants turn out to be low lifes.
Unfortunately, nothing you can do about his decision. If you can't get him to change his mind you'll have to move.

