Cal. CCP § 1176

(a) An appeal taken by the defendant shall not automatically stay proceedings upon the judgment. Petition for stay of the judgment pending appeal shall first be directed to the judge before whom it was rendered. Stay of judgment shall be granted when the court finds that the moving party will suffer extreme hardship in the absence of a stay and that the nonmoving party will not be irreparably injured by its issuance. If the stay is denied by the trial court, the defendant may forthwith file a petition for an extraordinary writ with the appropriate appeals court. If the trial or appellate court stays enforcement of the judgment, the court may condition the stay on whatever conditions the court deems just, but in any case it shall order the payment of the reasonable monthly rental value to the court monthly in advance as rent would otherwise become due as a condition of issuing the stay of enforcement. As used in this subdivision, “reasonable rental value” means the contract rent unless the rental value has been modified by the trial court in which case that modified rental value shall be used.

(b) A new cause of action on the same agreement for the rental of real property shall not be barred because of an appeal by any party.