  We study the decay rate of large deviation probabilities of occupation times,
up to time $t$, for the voter model $\eta\colon\Z^2\times[0,\infty)\ra\{0,1\}$
with simple random walk transition kernel, starting from a Bernoulli product
distribution with density $\rho\in(0,1)$. Bramson, Cox and Griffeath (1988)
showed that the decay rate order lies in $[\log(t),\log^2(t)]$.
  In this paper, we establish the true decay rates depending on the level. We
show that the decay rates are $\log^2(t)$ when the deviation from $\rho$ is
maximal (i.e., $\eta\equiv 0$ or 1), and $\log(t)$ in all other situations.
