We discuss how the modifications of the standard Einstein's equations needed to support the cosmological bounce can at the same time lead to vanishing of the well known cosmological constant problem, while also studying the effects of spacetime fluctuations. We first concentrate on the case of the classical FRWL spacetime and show that in a bouncing cosmology, where the modification of the Einstein-Hilbert action can be represented by the most general function needed to support the bounce, the cosmological constant problem is absent if this function at late cosmological times approaches a sufficiently large value. We show that this result is general and does not depend on the details of modifications needed to support the cosmological bounce. Therefore, the bouncing cosmologies could potentially at the same time solve the problem of singularity and cosmological constant. Motivated by the recent studies of cosmological constant problem in the context of fluctuations of the metric, we then generalize our study to incorporate the effects of spacetime fluctuations. We show that some problems of the recent earlier proposals, like singularities and negative values of the scale factor, are also naturally resolved in the approach proposed here.

32 pages, 9 figures