  We investigate the f(R) theory of gravity with broken diffeomorphism due to
the change of the coefficient in front of the total divergence term in the
(3+1)-decomposition of the scalar curvature. We perform the canonical analysis
of this theory and show that its consistent, i.e. with no unphysical degrees of
freedom, form is equivalent to the low-energy limit of the non-projectable f(R)
Horava-Lifshitz theory of gravity. We also analyze its cosmological solutions
and show that the de Sitter solution can be obtained also in the case of this
broken symmetry. The consequences of the proposed theory on the asymptotic
solutions of a few specific models in the cosmological context are also
presented.
