The answer is "maybe."
California Family Code Section 4323 says that there is a "rebuttable presumption, affecting the burden of proof, of decreased need for spousal support if the supported person is cohabitating with a person of the opposite sex."
This means if your supported ex takes up residence with a new boy or girlfriend the court assumes that support is no longer needed, unless your ex can prove otherwise. He or she has to prove the need for spousal support continues. You do not have to prove that he or she does not need it. The burden of proof is on the cohabitating supported party.
Like other spousal support modifications, the termination of support is not automatic. You need to bring a motion and allow the judge to hear all the facts. If you have further questions, please call my office for a no charge consultation.