The solution of a modified Hamilton–Jacobi equation with Lorentz-violating scalar field

A modified Hamilton–Jacobi equation with Lorentz-violating scalar field in the curved spacetime is used to research tunneling radiation from the horizons of Reissner–Nordstr o ¨ m de Sitter black hole. We investigate the influence of different etheric-like vector u α on the solution of the modified Hamilton–Jacobi equation by using both the semi-classical approximation and beyond the semi-classical approximation. We find that the solution of the modified Hamilton–Jacobi equation strongly depends on the choices of u α . As u α = δ t α u t , the Lorentz-violating scalar field can enhances black holes’ Hawking temperature but reduces its entropy. However, the case will reverses as u α = δ r α u r . Our conclusions are applicable to other spherically symmetric charged black holes.