Abstract of the talk// Oleksii presents a density functional-based equation of state for warm, dense nuclear matter with a transition to deconfined quark matter for applications to simulations of supernova explosions and neutron star mergers. The quark matter is modeled within a recently developed confining density functional approach, which provides suppression of quark degrees of freedom in the confining region by large values of their self-energy. He will compare the characteristics of the deconfinement phase transition modeled within the Maxwell construction scheme under the condition of isospin symmetry to the case of charge neutrality and beta equilibrium typical for protoneutron stars. In his talk he will also account for color-superconductivity of paired quark matter and show that it significantly affects the deconfinement phase transition and observational properties of neutron stars. Particularly, he shows that compared to unpaired quark matter, color-superconductivity lowers the onset density of quark deconfinement and reduces the density jump across the phase transition. To show how this impacts the mass-radius relation of neutron stars with quark cores he and other collaborating scientists analyzed the solutions of the equations of relativistic hydrostatics of these astrophysical objects. In the absence of color superconductivity, increasing entropy per baryon leads to the appearance of gravitational instability and formation of a disconnected third family of the neutron star sequence, twin stars. In the case of color-superconducting quark matter such disconnected neutron star sequence is absent. Finally, he will discuss the relation of the existence of thermal twin stars to the explodability of deconfinement-driven supernovae.