GUADALUPE, Ariz. — Although Joe Arpaio calls himself “America’s Toughest Sheriff,” a growing chorus of local critics want another title for him: Retired.

Sheriff Arpaio, the top law enforcement official in sprawling Maricopa County, is perhaps best known for his hard-nosed treatment of prisoners and his aggressive raids aimed at illegal immigrants. But it is his department’s approach to more than 400 sex-crimes cases that has Sheriff Arpaio in trouble.

His deputies failed to investigate or conducted only the sketchiest of inquiries into hundreds of sex crimes between 2005 and 2007, investigations by Arizona law enforcement agencies have shown. Many of those cases involved molested children.

The cases were first raised by The East Valley Tribune in 2008 but resurfaced in the news media earlier this year and in a recent article by The Associated Press, which prompted Sheriff Arpaio to defend himself at a news conference. “If there were any victims, I apologize to those victims,” he said on Monday, vowing to hold deputies accountable.