  An exact Kerr-like solution has been obtained recently in Einstein-bumblebee
gravity model where Lorentz symmetry is spontaneously broken. In this paper, we
investigate the superradiant instability of the Kerr-like black hole under the
perturbation of a massive scalar field. We find the Lorentz breaking parameter
$L$ does not affect the superradiance regime or the regime of the bound states.
However, since $L$ appears in the metric and its effect cannot be erased by
redefining the rotation parameter $\tilde{a}=\sqrt{1+L}a$, it indeed affects
the bound state spectrum and the superradiance. We calculate the bound state
spectrum via the continued-fraction method and show the influence of $L$ on the
maximum binding energy and the damping rate. The superradiant instability could
occur since the superradiance condition and the bound state condition could be
both satisfied. Compared with Kerr black hole, the nature of the superradiant
instability of this black hole depends non-monotonously not only on the
rotation parameter of the black hole $\tilde{a}$ and the product of the black
hole mass $M$ and the field mass $\mu$, but also on the Lorentz breaking
parameter $L$. Through the Monte Carlo method, we find that for $l=m=1$ state
the most unstable mode occurs at $L=-0.79637$, $\tilde{a}/M=0.99884$ and
$M\mu=0.43920$, with the maximum growth rate of the field
$\omega_{I}M=1.676\times10^{-6}$, which is about 10 times of that in Kerr black
hole.

General Relativity and Quantum Cosmology;High Energy Astrophysical Phenomena;High Energy Physics - Theory