  This work focuses on the temporal average of the backward Euler--Maruyama
(BEM) method, which is used to approximate the ergodic limit of stochastic
ordinary differential equations with super-linearly growing drift coefficients.
We give the central limit theorem (CLT) of the temporal average, which
characterizes the asymptotics in distribution of the temporal average. When the
deviation order is smaller than the optimal strong order, we directly derive
the CLT of the temporal average through that of original equations and the
uniform strong order of the BEM method. For the case that the deviation order
equals to the optimal strong order, the CLT is established via the Poisson
equation associated with the generator of original equations. Numerical
experiments are performed to illustrate the theoretical results.
