[Solved]: Is it $mathsf{NP}$-hard to decide whether $mathsf{P}=mathsf{NP}$?

A similar question is this:

Is $(a vee neg b) wedge (neg a vee b)$ $mathsf{NP}$-hard?

The question itself doesn’t make much sense, as $mathsf{NP}$ (as well as $mathsf{NP}$-hard) defines a set of languages, but $(a vee neg b) wedge (neg a vee b)$ is (just like $mathsf{P} = mathsf{NP}$) a word, an element of a language, not a language itself. But all in all… one can say that it is generally hard to prove (or disprove) $mathsf{P} = mathsf{NP}$, it might even be impossible (as discussed in the answer by Jernej).