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It is commonly recognized now that Dark Energy (or Lambda-term) is of crucial importance both at the early (inflationary) stage of cosmological evolution and at the present time. However, little is known about its nature and origin till now. In particular, it is still unclear if the Lambda-term is a new fundamental constant or represents just an effective contribution from the underlying field theory. Here, we show that a quite promising and universal approach to this problem might be based on the Mandelstam-Tamm uncertainty relation of quantum mechanics. Following this idea, we propose a cosmological model where the effective Lambda-term is important throughout the entire history of the Universe and decays gradually from the inflationary period to nowadays. The corresponding scale factor exhibits a "quasi-exponential" behavior (namely, proportional to the exponent of the square root of time). Besides, such an approach requires a substantial reconsideration of some other cosmological parameters, first of all, the age of the Universe. The current observational constraints and the opportunities for future verification of this model will be briefly outlined.