The court dismissed the request from Andrew Cook, saying the initial petition was not filed in the correct court. Cook's execution is set for 7 p.m.
Lawyers for Cook and Warren Lee Hill, another inmate set for execution this week, had filed a joint legal challenge arguing the state is violating the law by using pentobarbital in executions without a prescription. The state has said the challenge is frivolous.
Cook was convicted in the 1995 shooting deaths of two college students. Authorities say the killings were random.
Hill's execution has been put on hold by federal courts on a separate issue.