  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.
