The justices Monday refused to review a petition from 52-year-old Randall Wayne Mays. The former welder and oilfield worker was condemned for fatally shooting Henderson County Sheriff's Deputy Tony Ogburn in 2007. Another deputy, Paul Habelt, also was killed.
The deputies responded to a call of gunshots on Mays' property in Payne Springs, about 50 miles southeast of Dallas. After appearing to cooperate, Mays barricaded himself in his house and opened fire with a high-powered rifle.
An attorney argued Mays had poor legal help and was mentally impaired.
Mays does not have an execution date and has other appeals.