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We generalize the previously proposed running vacuum energy model byincluding a term proportional to \dot{H}, in addition to the existing H^2 term.We show that the added degree of freedom is very constrained if both lowredshift and high redshift data are taken into account. Best-fit models areundistinguishable from LCDM at the present time, but could be distinguished inthe future with very accurate data at both low and high redshifts. We stressthe formal analogy at the phenomenological level of the running vacuum modelswith recently proposed dark energy models based on the holographic or entropicpoint of view, where a combination of \dot{H} and H^2 term is also present.However those particular entropic formulations which do not have a constantterm in the Friedmann equations are not viable. The presence of this term isnecessary in order to allow for a transition from a decelerated to anaccelerated expansion. In contrast, the running vacuum models, both theoriginal and the generalized one introduced here contain this constant term ina more natural way. Finally, important conceptual issues common to all thesemodels are emphasized.

Generalizing the running vacuum energy model and comparing with the entropic-force models