We investigate the influence of the field-theoretical divergence on the thermal equilibrium property by employing the model of a uniformly accelerated monopole detector nonlinearly coupled to the scalar field in the flat space with the Dirichlet boundary. The ultraviolet singularity coming from the product of the field at the same spacetime point is renormalized by taking the normal-ordering procedure for the Minkowski vacuum in the interaction Lagrangian. We find that the deviation from the thermal equilibrium of the response of this detector is mostly due to interplay between the divergence and the existence of the boundary, when the detector is kept switched on during a large but finite amount of time. We discuss, in detail, how the thermal equilibrium property of the detector is recovered as the switch-on time goes to infinity.