“We don’t want to make it real comfortable for them because we don’t want them to want to come back,” Christopher Ivey, the chief sheriff’s deputy, said in his climate-controlled office two floors beneath the jail. “We try to get it and keep it at a level that it’s comfortable enough that they can survive.”

Mr. Ivey, who said no parish inmate had suffered a heat-related illness since the sheriff took office in 2012, said he believed the jail’s temperature never exceeded 80 degrees. But a jailer who dropped by Mr. Ivey’s office suggested that temperatures regularly reached the mid-90s.

“You don’t leave there not moist,” Mr. Ivey acknowledged. Parish officials, who have not faced a court challenge about jail temperatures, did not agree to requests for a tour of the facility or interviews with current inmates.

But after her release, Ms. Bourque, 25, described an environment where inmates found little relief.

“It’s hot as hell,” she said. “The church ladies come over there, and I told her that. And she was like, ‘No, I believe hell is hotter.’ And I was like, ‘It’s just an expression. It’s hot as hell.’”

Inmates, lawyers and doctors described similar conditions inside other jails across the South, and some said that temperatures endangered the lives of prisoners with health problems.

“Once these buildings heat up in the summertime, they never really do ever cool back down again,” Keith M. Cole, a plaintiff in the Texas class-action case, said at the Navasota prison where he is serving a life sentence for murder and is being treated for heart disease, diabetes and hypertension. “Air-conditioning to me wouldn’t be a comfort. It’s a necessity — it’s a medical necessity.”

Mr. Cole, 62, said that he understood public skepticism of air-conditioning for prisoners, and that he might have even embraced such an opinion before he was sentenced in 1995. But in an interview, he said, “This isn’t about comfort. This is about life or death.”