  The study of neutrino pair annihilation into electron-positron pairs
($\nu{\bar \nu}\to e^-e^+$) is astrophysically well-motivated because it is a
possible powering mechanism for the gamma-ray bursts (GRBs). In this paper, we
estimate the gamma-ray energy deposition rate (EDR) arising from the
annihilation of the neutrino pairs in the equatorial plane of a slowly rotating
black hole geometry modified by the broken Lorentz symmetry (induced by a
background bumblebee vector field). More specifically, owing to the presence of
a dimensionless Lorentz symmetry breaking (LSB) parameter $l$ arising from
nonminimal coupling between the bumblebee field with nonzero vacuum expectation
value and gravity, the metric solution in question differs from the standard
slowly rotating Kerr black hole. By idealizing the thin accretion disk
temperature profile in the two forms of isothermal and gradient around the
bumblebee gravity-based slow rotating black hole, we investigate the influence
of spontaneous LSB on the $\nu{\bar \nu}$-annihilation efficiency. For both
profiles, we find that positive values of LSB parameter $l>0$ induce an
enhancement of the EDR associated with the neutrino-antineutrino annihilation.
Therefore, the process of powering the GRBs jets around bumblebee gravity
modified slowly rotating geometry is more efficient in comparison with standard
metric. Using the observed gamma-ray luminosity associated with different GRBs
types (short, long, and ultra-long), we find, through the analysis of the EDR
in the parameter space $l-a$ ($a^2\ll1$), some allowed ranges for the LSB
parameter $l$.
