Writing in The Telegraph, Sir Robert said that he had changed his mind due to Ms Truss’s economic management, although that was one of the areas that led him to initially support Mr Sunak.
He described his switch as “not an easy thing to do” but said that he now believed he was endorsing “the right person to take our country forward”.
Ms Truss’ support for a British Bill of Rights, an attempt to move away from European human rights law, was also a factor for Sir Robert changing support.
Ms Truss remains consistently ahead in Tory membership polls to become the next prime minister.
Sir Robert wrote: “I have looked at ideas and principles first, and personalities second. During the parliamentary rounds, I backed Rishi Sunak as I felt that he was at that stage embodying what we needed.”
During the parliamentary stage, Mr Sunak was the frontrunner, but has failed to garner similar support from Tory members.
“As the campaign has moved on, and as I have listened carefully to both candidates, I have thought deeply about the issues that move me and what I want to see the next prime minister doing,” wrote Sir Robert.
“Changing your mind on an issue like this is not an easy thing to do, but I have decided that Liz Truss is the right person to take our country forward.”
He added: “Her plans give us our best shot at reaching our potential with the high-growth, high-productivity economy that we need not only to get us out of this crisis but to protect ourselves from the next.”
Sir Robert is the second Tory MP to publicly switch support from Mr Sunak to Ms Truss.
Chris Skidmore, the MP for Kingswood, also shared his support for Ms Truss in a Telegraph article, accusing Mr Sunak of repeatedly changing his position on key policy issues.