Legal Question in Landlord & Tenant Law in California

Yes, it the demand is made at the start of the rental period and thus is not changing an existing lease. It is not an unreasonable request since it reduces the potential liability the landlord faces and the likelihood of recovering if you damage something.

Mr. Shers is right but not exactly clear. For a term lease, such as six months or a year, the landlord cannot require or change any terms of the tenancy except when the lease is first signed or it is up for renewal. For a month-to-month rental agreement, or for a lease that has expired and rolled over to a month-to-month tenancy, the landlord can change the terms at any time upon 30-days notice. The demand and requirement that you carry the insurance is completely legal, not to mention the fact that you would be foolish not to have it anyway.