Latent voter model was considered in the mean-field setup by Lambiotte et al. We analyze this model on random regular graphs. In this model, each vertex of the underlying graph has a voter who has one of the two opinions ) 0 and 1, and is either active or inactive. When a voter is active, it adopts the opinion of an uniformly chosen neighbor at rate 1, and if this causes to changes its opinion, it enters the inactive phase. A voter stays inactive for an independent exponentially distributed amount of time with mean $1/\lambda$, during which it doesn't change its opinion. When $\lambda$ is large, although the model is a small perturbation of the ordinary voter model, the behavior changes discontinuously as the quasi-stationary density of each opinion tends to 1/2 irrespective of any positive starting level. The idea can be used to analyze many nonlinear voter models on locally tree-like random graphs.