In the problems of invariant distribution and density estimation of an ergodic diffusion process, the asymptotic variances of many estimators can be represented as some mathematical expectations with respect to the invariant law. Therefore, the construction of the confidence intervals requires the estimates of these quantities. The present work proposes estimators of them which are consistent and so it allows to construct the corresponding confidence intervals. Moreover, these quantities play an important role in the problems of asymptotically efficient estimation of the distribution function and density.