  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 exactly black hole solution and cosmological solutions in four
dimensional spacetime by a regularization scheme. The bumblebee field doesn't
affect the locations of the black hole horizon. It affects the gravitational
potential, decreases the black hole's Hawking temperature. This black hole is
different from Schwarzschild black hole that the gravitational potential has a
minimum in the black hole interior and is positive finite at short distance.
