Proving both the problem and the duty
Document two things: that the AC is broken, and that the landlord was responsible for it. Keep dated photos and videos, indoor temperature readings, your written repair requests, and proof the AC came with the rental — like the lease or a listing that advertised it.
Because broken AC only creates a duty in certain situations, showing the landlord was responsible is key. Save the lease language or ad that mentioned air conditioning, photos of the non-working unit, and a log of indoor temperatures over time.
Keep every text and email with your landlord, since these show what the landlord knew and when. If the AC leaked or caused mold, photograph that too, and hold on to any repair records. A clear record of both the problem and the landlord's duty tells the story. A free review can help you understand which evidence matters most in your situation.
This is general information about California tenant rights, not legal advice. Every situation is different.