Abstract Hořava gravity breaks Lorentz symmetry by introducing a dynamical timelike scalar field (the khronon), which can be used as a preferred time coordinate (thus selecting a preferred space–time foliation). Adopting the khronon as the time coordinate, the theory is invariant only under time reparametrizations and spatial diffeomorphisms. In the infrared limit, this theory is sometimes referred to as khronometric theory. Here, we explicitly construct a generalization of khronometric theory, which avoids the propagation of Ostrogradski modes as a result of a suitable degeneracy condition (although stability of the latter under radiative corrections remains an open question). While this new theory does not have a general-relativistic limit and does not yield a Friedmann–Robertson–Walker-like cosmology on large scales, it still passes, for suitable choices of its coupling constants, local tests on Earth and in the Solar System, as well as gravitational-wave tests. We also comment on the possible usefulness of this theory as a toy model of quantum gravity, as it could be completed in the ultraviolet into a ‘degenerate Hořava gravity’ theory that could be perturbatively renormalizable without imposing any projectability condition.