Deciding whether the language of a CFG equals some specific regular language I was wandering if $L = \{\langle G \rangle \mid \ G \text{ is a context free grammar and } \mathcal L(G) = A \}$ is decidable where A is a some regular language. Is 
$L' = \{ \langle G \rangle \mid \ G \text{ is a context free grammar and } \mathcal L(G) = \overline A \}$ decidable ?  I feel like $L'$ should be decidable, but can't see how. Or does it depends on $A$?
It depends on your choice of $A$. For example, the problem $\{\langle G \rangle \mid L(G) = \Sigma^* \}$ is undecidable, while $\{\langle G \rangle \mid L(G) = \phi \}$ is decidable.

Also, the two languages that you have given (the one with $A$ and the other with $\overline{A}$) are equivalent since regular languages are closed under complementation.