Whenever any prisoner is taken from jail, or from the custody of any sheriff or his deputy, and put to death, or suffers grievous bodily harm, owing to the neglect, connivance, cowardice, or other grave fault of the sheriff, such sheriff may be impeached...
Ala. Const. § 138.
Yep, in Alabama, sheriffs can be impeached for cowardice, if a prisoner gets lynched.
Oh, and since it also seems interesting, the governor and other high state officials can be impeached for "intemperance in the use of intoxicating liquors..." Ala. Const. § 173.