  The behaviour of a relativistic scalar particle subject to a scalar potential
under the effects of the violation of the Lorentz symmetry in the cosmic string
spacetime is discussed. It is considered two possible scenarios of the Lorentz
symmetry breaking in the CPT-even gauge sector of the Standard Model Extension
defined by a tensor $\left(K_{F}\right)_{\mu\nu\alpha\beta}$. Then, by
introducing a scalar potential as a modification of the mass term of the
Klein-Gordon equation, it is shown that the Klein-Gordon equation in the cosmic
string spacetime is modified by the effects of the Lorentz symmetry violation
backgrounds and bound state solution to the Klein-Gordon equation can be
obtained.
