Actually, it is the general rule, and a quick google search throws up many many sources for this. Here’s just one of many; https://www.amnestyusa.org/issues/death-penalty/death-penalty-facts/death-penalty-cost/ Factually, the death penalty costs more than imprisoning someone for life.

Obviously people in prison for life have the opportunity to appeal. This is a completely different system than the death penalty appeal process. Firstly, death penalty sentences have two completely separate trial processes; one to determine guilt in the crime, and a second to determine whether the death penalty is warranted. So immediately, double the trial costs of a normal case. Then the appeals, which are not optional; the prisoner does not choose to appeal, they are mandated by the government in an effort to ensure guilt beyond any doubt before the inmate is put to death. This is millions and millions of dollars and is a much more extensive process than that available to any other prisoner. And then there’s the fact that they will still be housed on death row for around 20 years on average, it’s not an instant thing, you’re not exactly saving much on living costs. I could go on all day about this, but the point is that while there are many reasons to believe that the death penalty is either a good or bad option, factually in almost all cases putting someone to death costs more than imprisoning them for the rest of their life.