A type of exponential correction to General Relativity gives viable modified gravity model of dark energy. The model behaves as $R-2\Lambda$ at large curvature where an effective cosmological constant appears, but it becomes zero in flat space time. The cosmic evolution of the main density parameters is consistent with current observations. The thin shell conditions for the Solar system were analyzed. Apart from satisfying cosmological and local gravity restrictions, the model may also show measurable differences with $\Lambda$CDM at recent times. The current value of the deviation parameter $m$ for scales relevant to the matter power spectrum can be larger than $10^{-6}$. The growth index of matter density perturbations is clearly different from that of the $\Lambda$CDM. The theoretical predictions of the model for the weighted growth rate were analyzed in the light of the $f\sigma_8$-tension.

Comment: 39 pages, 9 figures. version published in CQG