In the author’s model of low-energy quantum gravity, the gravitational interaction of bodies arises as the effect of screening in a sea of superstrongly interacting gravitons. Newton's constant can be calculated as a statistical parameter of the repeated interaction of bodies with gravitons. Small vacuum effects in the model, caused by a new type of interaction -photon-graviton - may have major implications for cosmology. The ones are reviewed here and compared with cosmological observations; this is a self-review of author's works in this approach. The constancy of the ratio $H(z)/(1+z)$ in this model is consistent with observations of the Hubble parameter $H(z)$.Cosmological redshift is a local quantum effect caused by head-on collisions of photons with gravitons, while the additional dimmingof distant objects and diffuse cosmic optical background supposedly detected by the New Horizons mission are due to non-head-on collisions. For very soft radiation, the additional relaxation factor is calculated. Since this factor must have different values for soft and hard radiation, the distance modulusin this model must be a multi-valued function of the redshift. The Hubble tension is discussed in this context. The considered quantum effects are beyond the scope of the standard cosmologicalmodel. These small effects can describe cosmological observations in a very elegant and unified manner without dark energy andcosmological expansion. Graviton-graviton interactions can lead to the formation of virtual gravitons with very low mass, which maybe candidates for the role of dark matter particles.