Einstein-Gauss-Bonnet gravity coupled to bumblebee field in four dimensional spacetime

We study Einstein-Gauss-Bonnet gravity coupled to a bumblebee field which leads to a spontaneous Lorentz symmetry breaking in the gravitational sector. We obtain an exact black hole solution and a cosmological solution in four dimensional spacetime by a regularization scheme. We also obtain a Schwarzschild-like bumblebee black hole solution in $D$-dimensional spacetime. We find that the bumblebee field doesn't affect the locations of the black hole horizon, but only affects the gravitational potential. That is, its gravitational potential has a minimum value(negative) in the black hole interior and has a positive value $1+\ell$ at short distance $r\rightarrow0$. If the constant $\ell$ is large enough, then this kind of black hole is practically free from the singularity problem. The thermodynamics and phase transition are also studied. In a cosmological context, it is interesting that the Gauss-Bonnet term has no effect on the conservation of energy equation. A late-time expansion of de Sitter universe can be replicated in an empty space. The Gauss-Bonnet term and the bumblebee field can both actually act as a form of dark energy.