We reexamine the classic problem of the renormalization of zero-point quantum fluctuations in a Friedmann-Robertson-Walker background. We discuss a number of issues that arise when regularizing the theory with a momentum-space cutoff, and show explicitly how introducing noncovariant counterterms allows to obtain covariant results for the renormalized vacuum energy-momentum tensor. We clarify some confusion in the literature concerning the equation of state of vacuum fluctuations. Further, we point out that the general structure of the effective action becomes richer if the theory contains a scalar field ϕ with mass m smaller than the Hubble parameter H(t). Such an ultralight particle cannot be integrated out completely to get the effective action. Apart from the volume term and the Einstein-Hilbert term, that are reabsorbed into renormalizations of the cosmological constant and Newton's constant, the effective action in general also has a term proportional to F(ϕ)R, for some function F(ϕ). As a result, vacuum fluctuations of ultralight scalar fields naturally lead to models where the dark energy density has the form ρDE(t) =ρx(t) + ρz(t), where ρX is the component that accelerates the Hubble expansion at late times and ρz(t)is an extra contribution proportional to H2(t). We perform a detailed comparison of such models with CMB, SNIa and BAO data