  We investigate static and spherically symmetric solutions in a gravity theory
that extends the standard Hilbert-Einstein action with a Lagrangian constructed
from a three-form field $A_{\alpha \beta \gamma}$, which is related to the
field strength and a potential term. The field equations are derived from a
variational principle and are obtained explicitly for a static and spherically
symmetric geometry in vacuum. For the case of the vanishing three-form field
potential the gravitational field equations can be solved exactly. However, for
arbitrary potentials, due to their mathematical complexity, numerical
approaches are adopted in studying the behavior of the metric functions and the
three-form field. To this effect, the field equations are reformulated in a
dimensionless form and are solved numerically by introducing a suitable
independent radial coordinate. We detect the formation of a black hole from the
presence of a Killing horizon for the time-like Killing vector in the metric
tensor components. Several models, corresponding to different functional forms
of the three-field potential, namely, the Higgs and exponential type, are
considered. In particular, naked singularity solutions are also obtained for
the exponential potential case. Finally, the thermodynamic properties of these
black hole solutions, such as the horizon temperature, specific heat, entropy
and evaporation time due to the Hawking luminosity, are also investigated in
detail.
