Does an unspecified cosmological constant solve the problem of time in quantum gravity?

In unimodular gravity, an unspecified cosmological constant appears as a variable canonically conjugate to a four-volume variable, the cosmological time. It was suggested that this time sets the conditions of quantum measurements and solves thereby the interpretation problems of quantum geometrodynamics. By analyzing the relationship of the cosmological time to hypertime (the collection of spacelike hypersurfaces), we highlight the difficulties of such a position. The constraint system of parametrized unimodular gravity implies that the cosmological time labels only equivalence classes formed by hypersurfaces separated by a zero four-volume, while individual spacelike hypersurfaces within an equivalence class are physically irrelevant. As a result, unless complemented by a hypertime variable, cosmological time does not uniquely set the conditions for measuring geometric variables either in the classical or in the quantum theory.