  We analyze the thermodynamical properties of black holes in a modified theory
of gravity, which was initially proposed to obtain correct dynamics of galaxies
and galaxy clusters without dark matter. The thermodynamics of non-rotating and
rotating black hole solutions resembles similar solutions in Einstein-Maxwell
theory with the electric charge being replaced by a new mass dependent
gravitational charge $Q = \sqrt{\alpha G_N}M$. This new mass dependent charge
modifies the effective Newtonian constant from $G_N$ to $G = G_N(1+\alpha)$,
and this in turn critically affects the thermodynamics of the black holes. We
also investigate the thermodynamics of regular solutions, and explore the
limiting case when no horizons forms. So, it is possible that the modified
gravity can lead to the absence of black hole horizons in our universe.
Finally, we analyze corrections to the thermodynamics of a non-rotating black
hole and obtain the usual logarithmic correction term.
