It's an interesting question, copyright doesn't cover tone/pitch/etc... so how do you protect that? Should you even be allowed to?
So you could still deep fake voice actors with the intention of just saving money rather than wanting people to recognise them.
I'd go much farther than that. Unique features of someone's voice and appearance (face, walk) should not only be protected by copyright but also by privacy laws. It would be crazy if producers were allowed to make arbitrary videos with anyone's appearance and voice, putting words in your mouth you'd never endorse.
Luckily, this may be one of the cases when politics will react fast and swiftly, if necessary with new laws, since politicians would likely be among the first victims of the new trend.