  The Kalb-Ramond (KR) gravity theory, a modified gravity theory that
nonminimally couples a KR field with a nonzero vacuum expectation value for the
gravitational field, can spontaneously break the Lorentz symmetry of gravity.
In a recent work, Yang et al. [Phys. Rev. D 108, 124004 (2023)] successfully
derived Schwarzschild-like black hole solutions both with and without a nonzero
cosmological constant within the framework of KR gravity. However, their
analysis did not address the more general case of static, neutral, spherically
symmetric black holes. In this paper, we fill this gap by resolving the field
equations to construct more general static, neutral, spherically symmetric
black hole solutions both with and without a nonzero cosmological constant. Our
black hole solutions are shown to obey the first law and the Bekenstein-Smarr
mass formulas of black hole thermodynamics. Moreover, we demonstrate that our
static neutral spherically symmetric AdS black hole does not always satisfy the
reverse isoperimetric inequality (RII), as the isoperimetric ratio can be
larger or smaller than unity depending on the placement of the solution
parameters within the parameter space. This behavior contrasts with the
above-mentioned Schwarzschild-like AdS black hole in the KR gravity theory,
which always obeys the RII. Significantly, the present more general static,
neutral, spherically symmetric AdS black hole is the first example of a static
AdS black hole that can violate the RII.
