  We consider a modified gravity model which we call "dynamical
Henneaux-Teitelboim gravity" because of its close relationship with the
Henneaux-Teitelboim formulation of unimodular gravity. The latter is a fully
diffeomorphism-invariant formulation of unimodular gravity, where full
diffeomorphism invariance is achieved by introducing two additional
non-dynamical fields: a scalar, which plays the role of a cosmological
constant, and a three-form whose exterior derivative is the spacetime volume
element. Dynamical Henneaux-Teitelboim gravity is a generalization of this
model that includes kinetic terms for both the scalar and the three-form with
arbitrary couplings. We study the field equations for the cases of spherically
symmetric and homogeneous, isotropic configurations. In the spherically
symmetric case, we solve the field equations analytically for small values of
the coupling to obtain an approximate black hole solution. In the homogeneous
and isotropic case, we perturb around de Sitter space to find an approximate
cosmological background for our model.
