is the basic actor-critic approach from the Sutton book only for continuous action spaces?

Question

I went through some literature and I tried to implement the Sutton & Barto basic actor critic net with entropy and TD Error on a big discrete actionspace. But it is not performing so well.

So I checked some forums, and it sad, that for discrete action spaces, I need to implement an adaption, e.g. the Soft-Actor Critic (SAC). Is that true? I couldn't find a scientific paper where it said something like that?

Thank you for your help