Revisiting a parity violating gravity model without ghost instability: Local Lorentz covariance

Recently, based on the theory of teleparallel gravity, a simple and ghost free parity violating gravity model was proposed in [M. Li, H. Rao, D. Zhao, J. Cosmol. Astropart. Phys. 11 (2020) 023], where Weitzenb\"{o}ck connection adopted for simplifying calculations but breaks local Lorentz symmetry explicitly. In this paper, we restore by giving up condition spin connection. With full covariance, is not pure tetrad any more. We also apply new version to universe with general Friedmann-Robertson-Walker background. This further generalizes studies 023] only spatially flat background considered. Through investigations confirm results obtained addition get some results.