Hart, sacked himself by Nottingham Forest last month, has agreed a rolling contract at Barnsley.
Thordarson was dismissed following a poor run of just one win in 13 games, including last week’s 6-1 thrashing at Grimsby.
Hart, who was linked with Leeds after Peter Reid's sacking, first game in charge will be at home to Luton.
Tykes chairman Ridsdale said: "Any manager is always judged on results.
"One win in 13 games has meant a play-off spot, which for the first half this season looked a distinct possibility, now looks less certain."