At the end of season 5 episode 18 when Flash went to Iron Height prison to see Eobard Thawne in future, Thawne told him to forgive Nora and to consider it a condemned man's last request. In the next episode, Iris told Barry that Thawne is in Death row. We can't say this for sure yet, But it is safe to assume that the countdown is for Eobard Thawne's execution. Update: After Season 5 episode 21 we have confirmation that Eobard Thawne is in death row, he is supposed to be executed when the countdown hits 00:00:00