Using Monte Carlo methods, we study $Z_2$ symmetry in $Z_2+$Higgs theory. In $3+1$ space-time dimensions, our simulation results suggest that the $Z_2$ symmetry is realized at large number of temporal lattice points ($N_\tau$) in the Higgs symmetric phase. In order to see the dependence of $Z_2$ symmetry on the number of temporal lattice points we have also studied a simple temporal one dimensional model for a given spatial site. We show that the $Z_2$ symmetry is observed at the level of free energy at large $N_\tau$ limit. For this model, we also compute the density of states (DoS) for various $N_\tau$ values, where we show that the realization of the $Z_2$ symmetry happens at higher $N_\tau$. Therefore, the realization of $Z_2$ symmetry may be due to dominance of the DoS.