On cohomological $C^0$-(in)stability

After Katok, a homeomorphism $f\text{\hspace{0.17em}⁣}:M\to M$ is said to be cohomologically $C^0$-stable when its space of real $C^0$-coboundaries is closed in $C^0(M)$. In this short note we completely classify cohomologically $C^0$-stable homeomorphisms, showing that periodic homeomorphisms are the only ones.