Using data collected with the D0 detector at the Fermilab Tevatron Collider, corresponding to $5.3\text{ }\text{ }{\mathrm{fb}}^{\ensuremath{-}1}$ of integrated luminosity, we search for violation of Lorentz invariance by examining the $t\overline{t}$ production cross section in $\mathrm{\text{lepton}}+\mathrm{\text{jets}}$ final states. We quantify this violation using the standard-model extension framework, which predicts a dependence of the $t\overline{t}$ production cross section on sidereal time as the orientation of the detector changes with the rotation of the Earth. Within this framework, we measure components of the matrices $({c}_{Q}{)}_{\ensuremath{\mu}\ensuremath{\nu}33}$ and $({c}_{U}{)}_{\ensuremath{\mu}\ensuremath{\nu}33}$ containing coefficients used to parametrize violation of Lorentz invariance in the top quark sector. Within uncertainties, these coefficients are found to be consistent with zero.