  As a vector-tensor theory including nonminimal coupling between the Ricci
tensor and a vector field, the bumblebee gravity is a potential theory to test
Lorentz symmetry violation. Recently, a new class of numerical spherical black
holes in the bumblebee theory was constructed. In this paper, we investigate
the associated local thermodynamic properties. By introducing a pair of
conjugated thermodynamic quantities $X$ and $Y$, which can be interpreted as an
extension of electric potential and charge of the Reissner Nordstr\"om black
holes, we numerically construct a new first law of thermodynamics for bumblebee
black holes. We then study the constant-$Y$ processes in the entropy-charge
parameter space. For the constant-$Y$ processes, we also calculate the heat
capacity to study the local thermodynamic stability of the bumblebee black
holes. For a negative nonminimal coupling coefficient $\xi$, we find both
divergent and smooth phase transitions. For a positive but small $\xi$, only a
divergent phase transition is found. It turns out that there is a critical
value $0.4\kappa <\xi_c < 0.5\kappa$ such that when $\xi_c < \xi<2\kappa$, even
the divergent phase transition disappears and the bumblebee black holes thus
become locally thermodynamically unstable regardless of the bumblebee charge.
As for $\xi>2\kappa$, the smooth phase transition arises again but there no
longer exists any discontinuous phase transition for the bumblebee black holes.
