The high court's 60-page opinion ruled against the appeal of Richard Baumhammers, currently on death row for the two-hour rampage. He also shot a sixth man who was left paralyzed and has since died.
Baumhammers began by shooting to death a Jewish neighbor and setting her house on fire, then attacked others at a Chinese restaurant and Indian grocery. He painted racist graffiti and fired into two synagogues.
Nothing in his argument, Justice Thomas Saylor for the unanimous court, "convinces us that he is entitled to relief."
The appeal addressed statements by a forensic psychiatrist after the trial, asserted mistakes by his defense lawyer at trial and sentencing, raised procedural claims and said Baumhammers was forcibly medicated at trial in violation of his rights.
A spokesman for the Allegheny County district attorney's office declined to comment on the decision, and Baumhammers' lawyer did not return a phone message.
Baumhammers, 49, is imprisoned in Greene State Prison in southwestern Pennsylvania.