Hawking tunneling radiation in Lorentz-violating scalar field theory

In this paper, the Lorentz-violating scalar field equation is generalized in curved spacetime, and we find that the aether-like terms modify the scalar field equation, so that the effect can correct the properties near the event horizon of black hole spacetime. We then obtain the modified Hamilton-Jacobi equation by semi-classical approximation, and investigate the Hawking radiation and black hole thermodynamics in Schwarzschild black hole spacetime. The results show that the effects of aether-like terms increase the temperature of black hole, but reduce the entropy of black hole as \begin{document}${{ u}^\alpha } = {\text{δ}}_t^\alpha {u^t}, {\text{δ}}_r^\alpha {u^r}$\end{document}. This work can help to understand the properties of Lorentz-violating in curved spacetime.