The situations that create a duty
Broken air conditioning can become a legal issue when the lease included AC and the landlord will not maintain it, when a local ordinance requires cooling, when extreme heat creates a real health or safety danger, or when the AC unit itself is leaking, growing mold, or unsafe.
Because state law does not require AC, the question is usually whether something else creates a duty. The most common is a lease that provided AC — landlords generally must maintain the amenities they agreed to supply.
Beyond that, a few cities have cooling rules, dangerous indoor heat can raise health and safety concerns (especially for infants, older adults, or people with medical conditions), a tenant with a disability may be able to request a cooling-related accommodation, and problems like an AC leaking water or posing an electrical hazard are habitability issues in their own right. If your broken AC involves one of these situations, a free review can help you understand your rights.
This is general information about California tenant rights, not legal advice. Every situation is different.