The duty to maintain a provided amenity
Generally, yes. A landlord who provided air conditioning with the rental is usually responsible for maintaining and repairing it, because landlords are typically expected to keep the amenities they supply in working condition.
This is the key point with AC. While a landlord may not have to add cooling to a unit that never had it, once they provide AC, they generally cannot let it fall into disrepair and simply refuse to fix it. That applies to a central system, a built-in wall unit, or a window unit that came with the rental.
A landlord who provided AC and then ignores repair requests may be failing a duty they took on themselves. Keep records showing the unit came with AC, such as the lease or the rental listing. If your landlord provided air conditioning and will not maintain it, a free review can help you understand your rights.
This is general information about California tenant rights, not legal advice. Every situation is different.