Casimir effect of a rough membrane in an Aether-like Lorentz-violating scenario

We explore the Casimir effect of a rough membrane within the framework of theories that break Lorentz symmetry. We consider two constant Aether vectors: one timelike and other spacelike, simultaneously. We employ an appropriate change of coordinates such that the membrane assumes a completely flat border and the remaining terms associated with the roughness are considered as part of the potential. Quantum fluctuations are induced by a scalar quantum field subject to Dirichlet boundary conditions. The spectrum is obtained through perturbation theory and regularized using the $\zeta$--function method. We provide an explicit example of a membrane with periodic boundaries. The presence of Aether vectors has a significant impact on the dominant term of the Casimir effect, while roughness only affects the secondary terms. Additionally, we examine the finite-temperature case.