Do we know eventually p(e)?

A quasi-particle model is employed to derive from available lattice QCD calculations an equation of state useable in hydrodynamical simulations of the expansion stage of strongly interacting matter created in ultra-relativistic heavy-ion collisions. Various lattice results give an astonishing agreement of the pressure as a function of energy density at large energy densities supposed the pseudo-critical temperature is in the range $170 \pm 15$ MeV, while in the transition region the equation of state is not yet well constrained. Therefore, one can construct a family of equations of state by bridging the uncertain region from the uniquely given high-energy density region part to a hadronic equation of state by suitable interpolation together with the extrapolation to non-zero baryon density by means of the quasi-particle model. We present a series of tests of the model, discuss the chiral extrapolation and the role of Landau damping. We also briefly sketch the path of cosmic matter in the early universe in the phase diagram.