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We continue the study of covariant power-counting renormalizable gravityconstrained by scalar Lagrange multiplier. Lorentz symmetry breaking isinvestigated in such a theory in comparison with the one in ghost condensationmodel. Covariant power-counting renormalizable vector gravity which isinvariant under $U(1)$ gauge symmetry is proposed. Several forms of vectorLagrange multiplier in this theory are discussed. It is shown that covariantscalar/vector gravity under consideration may have power-law or de Sitteraccelerating cosmological solution corresponding to inflationary era. Simplestblack hole solution is obtained and dispersion relations for graviton arepresented.

Covariant power-counting renormalizable gravity: Lorentz symmetry breaking and accelerating early-time FRW universe