Klein-Gordon oscillator subjected to aether-like Lorentz symmetry violation

We investigate bound-state solutions of the Klein--Gordon oscillator for spin-$0$ particles in the presence of aether-like Lorentz-symmetry violation. The Lorentz-violating effects are introduced through a nonminimal derivative coupling involving a fixed background vector field, and two physically distinct configurations are considered: a purely time-like background and a radial space-like background. In both cases, after separating the temporal, radial, and angular variables, the modified radial equation is reduced to the Whittaker equation. Normalizable bound-state solutions are then obtained by imposing the polynomial-termination condition on the associated confluent hypergeometric functions, leading to exact analytical expressions for the relativistic energy spectra. For the time-like configuration, the Lorentz-violating parameter enters the spectrum through a universal multiplicative factor. As a consequence, the complete positive-energy spectrum is globally rescaled, while the ordering of the levels and their relative structure are preserved. Positive values of the effective coupling lower the positive-energy levels, whereas negative values, subject to the reality condition of the spectrum, raise them. In contrast, the radial space-like background modifies the radial and angular contributions independently. The resulting corrections are explicitly dependent on the quantum numbers, and the same sign of the Lorentz-violating parameter may raise some levels while lowering others. We also derive weak-coupling expansions for both configurations and compare them with the exact spectra. The numerical analysis confirms the recovery of the Lorentz-symmetric Klein--Gordon oscillator in the vanishing-coupling limit, illustrates the qualitatively different spectral responses of the time-like and radial space-like backgrounds, and verifies the accuracy of the first-order approximations in the perturbative regime. These results provide a direct characterization of how aether-like Lorentz-symmetry violation can modify relativistic bound-state energies without destroying the analytical solvability of the Klein--Gordon oscillator.