  We consider the internal structure and the physical properties of specific
classes of neutron, quark and Bose-Einstein Condensate stars in the hybrid
metric-Palatini gravity theory, which is a combination of the metric and
Palatini $f(R)$ formalisms. The theory is very successful in accounting for the
observed phenomenology, since it unifies local constraints at the Solar System
level and the late-time cosmic acceleration, even if the scalar field is very
light. We derive the equilibrium equations for a spherically symmetric
configuration (mass continuity and Tolman-Oppenheimer-Volkoff) in the framework
of hybrid metric-Palatini theory, and we investigate their solutions
numerically for different equations of state of neutron and quark matter, by
adopting for the scalar field potential a Higgs-type form. Stellar models,
described by the stiff fluid, radiation-like, the bag model and the
Bose-Einstein Condensate equations of state are explicitly constructed in both
General Relativity and hybrid metric-Palatini gravity, thus allowing an in
depth comparison between the predictions of these two gravitational theories.
As a general result it turns out that for all the considered equations of
state, hybrid metric-Palatini gravity stars are more massive than their general
relativistic counterparts. Furthermore, two classes of stellar models
corresponding to fixed forms of the scalar field are also investigated.
Interestingly enough, in the case of a constant scalar field the equation of
state of the matter takes the form of the bag model equation of state
describing quark matter. As a possible astrophysical application of the
obtained results we suggest that stellar mass black holes, with masses in the
range of $3.8M_{\odot}$ and $6M_{\odot}$, respectively, could be in fact hybrid
metric-Palatini gravity neutron or quark stars.
