We construct a semiclassical Friedmann–Lemaître–Robertson–Walker (FLRW) cosmological model assuming a running cosmological constant (CC). It turns out that the CC becomes variable at arbitrarily low energies due to the remnant quantum effects of the heaviest particles, e.g., the Planck scale physics. These effects are universal in the sense that they lead to a low-energy structure common to a large class of high-energy theories. Remarkably, the uncertainty concerning the unknown high-energy dynamics is accumulated into a single parameter ν, such that the model has an essential predictive power. Future Type Ia supernovae experiments (like SNAP) can verify whether this framework is correct. For the flat FLRW case and a moderate value ν10⁻², we predict an increase of 10–20% in the value of Ω_Λ at redshifts z=1–1.5 perfectly reachable by SNAP.