It is a classical fact (e.g. here) that for an (accessible) right adjoint comonad on a (sheaf) topos, its Eilenberg-Moore category of coalgebras is itself a (sheaf) topos.

I suppose this remain true for $\infty$-toposes, for hypercomplete $\infty$-stack $\infty$-toposes at least? (But not assuming that the comonad is idempotent.)

It is a classical fact (e.g. here) that for an (accessible) right adjoint comonad on a (sheaf) topos, its Eilenberg-Moore category of coalgebras is itself a (sheaf) topos.

I suppose this remains true for $\infty$-toposes, for hypercomplete $\infty$-stack $\infty$-toposes at least? (But not assuming that the comonad is idempotent.)