A Texas court has halted the execution of a man convicted of killing his two-year-old daughter in a "shaken baby syndrome" case.

Death row inmate Robert Roberson, who has always maintained the child died from complications linked to pneumonia, had been scheduled for execution on 16 October.

The diagnosis of shaken baby syndrome refers to a serious brain injury resulting from forcefully shaking an infant or a toddler.

On Thursday, the Texas Court of Criminal Appeals granted an emergency stay under the state's 2013 "junk science" law that allows courts to re-examine convictions based on scientific evidence that has since been debunked or evolved.

In recent years, lawmakers from both major parties have urged a re-examination of the case, which Roberson's lawyers say was built on outdated medical theories.

It is not the first time Roberson's execution has been paused. He was set to be executed last October, but in a last-minute move, a bipartisan group of state lawmakers issued a subpoena for Roberson to testify in a hearing, pausing his execution as the warrant for it expired at midnight.

Texas Attorney General Ken Paxton has continued to back the execution, arguing that Roberson abused his daughter.

A post-mortem examination found the child died of injuries from abuse but Roberson maintains his daughter fell out of her bed and when he returned hours later she was not breathing.

Medical staff at the emergency room where he took her immediately suspected abuse.

Roberson's lawyers argued his daughter had been prescribed medicines that are no longer given to children because they can cause serious complications and that, along with the fall, was the cause of her death.

Brian Wharton, the lead detective on the case, is among those supporting a plea for releasing Roberson, writing in a letter that he "will be forever haunted by my participation in his arrest and prosecution. He is an innocent man".