A state law that prohibits local government entities from legislating on a particular topic(s) is known as a preemption law. The state “preempts” a local entity’s ability to legislate on that topic because the state believes that the topic is something that should be dealt with on a state-wide basis rather than locally.
Some states have passed preemption laws prohibiting local government entities from enacting an ordinance requiring employers to provide sick leave. As you would expect, this aspect of paid sick leave is dynamic as well. In some states, there are court challenges to the state’s ability to enact such a law. In others, bills have been introduced to overturn the preemption law. It is also quite possible, perhaps likely, that more states will enact such laws.
I have added a “Preemption States” tab to this blog’s menu bar. If you click on this tab, you will see a list of states that have enacted preemption laws as of September 30, 2016. Clicking on the state you are interested in will link you to the legislative source document for that state.