  We consider interior 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 generates,
via the field strength and a potential term, a new component in the total
energy-momentum tensor of the gravitational system. We formulate the field
equations in Schwarzschild coordinates and investigate their solutions
numerically for different equations of state of neutron and quark matter, by
assuming that the three field potential is either a constant or possesses a
Higgs-like form. Moreover, stellar models, described by the stiff fluid,
radiation-like, bag model and the Bose-Einstein condensate equations of state
are explicitly obtained in both general relativity and three-form gravity, thus
allowing an in-depth comparison between the astrophysical predictions of these
two gravitational theories. As a general result we find that for all the
considered equations of state, three-form field stars are more massive than
their general relativistic counterparts. As a possible astrophysical
application of the obtained results, we suggest that the 2.5$M_{\odot}$ mass
compact object, associated with the GW190814 gravitational wave event, could be
in fact a neutron or a quark star described by the three-form field gravity
theory.
