A new scheme for testing the nuclear matter (NM) equation of state (EoS) at high densities using constraints from compact star (CS) phenomenology is applied to neutron stars with a core of deconfined quark matter (QM). An acceptable EoS shall not to be in conflict with the mass measurement of 2.1 +/- 0.2 M_sun (1 sigma level) for PSR J0751+1807 and the mass-radius relation deduced from the thermal emission of RX J1856-3754. Further constraints for the state of matter in CS interiors come from temperature-age data for young, nearby objects. The CS cooling theory shall agree not only with these data, but also with the mass distribution inferred via population synthesis models as well as with LogN-LogS data. The scheme is applied to a set of hybrid EsoS with a phase transition to stiff, color superconducting QM which fulfills all above constraints and is constrained otherwise from NM saturation properties and flow data of heavy-ion collisions. We extrapolate our description to low temperatures and draw conclusions for the QCD phase diagram to be explored in heavy-ion collision experiments.