  Since the spontaneously broken of the Lorentz symmetry in the gravity theory
with the non-minimally coupling between the Kalb-Ramond (KR) field (that
acquires a nonzero vacuum expectation value) and the Einstein gravity, there
exists the exactly static and spherically symmetric black holes solutions
related with the Lorentz violating parameter. Based on this, we consider the
corresponding black hole solution in the de-Sitter (dS) spacetime with the KR
field and investigate the thermodynamic properties in the expanded phase space
through introducing the interplay entropy between the black hole and
cosmological horizons. Especially we analyze the effect of the
Lorentz-violating parameter on the thermodynamic properties. Furthermore, the
Lyapunov exponent and the shadow of these static and spherically symmetric
black holes in this Lorentz-violating gravity theory are also investigated.
These study will open a new perspective to probe the thermodynamics of black
holes.
