The full 11-judge Ninth Circuit Court of Appeals granted the stay so an appeal rejected by a three-judge panel Sunday night could be heard by the full court in San Francisco.
Cooper was convicted of the murders of two adults and two children in San Bernardino County in 1983. Cooper, who escaped from prison shortly before the murders, was believed hiding near the home of the victims at the time.
In his stay request, Cooper's lawyers argue the state should have tested DNA evidence at the first opportunity.
California officials say DNA testing links Cooper to the murders; although Cooper's advocates contend there are questions about forensic evidence and new witness statements that must be addressed before their client is put to death.
Notable deaths of 2014 [PHOTOS]