  This review covers several recent developments in the physics of dense QCD
with an emphasis on the impact of multiple phase transitions on astrophysical
manifestations of compact stars. It is conjectured that pair-correlated quark
matter in $\beta$-equilibrium is within the same universality class as
spin-imbalanced cold atoms and the isospin asymmetrical nucleonic matter. This
then implies the emergence of phases with broken space symmetries and
tri-critical (Lifshitz) points. We construct an equation of state (EoS) that
extends the two-phase EoS of dense quark matter within the constant speed of
sound parameterization by adding a conformal fluid with a speed of sound
$c_{\rm conf.}=1/\sqrt{3}$ at densities $\ge 10~n_{\rm sat}$, where $n_{\rm
sat}$ is the saturation density. With this input, we construct static,
spherically symmetrical compact hybrid stars in the mass--radius diagram,
recover such features as the twins and triplets, and show that the transition
to conformal fluid leads to the spiraling-in of the tracks in this diagram.
Stars on the spirals are classically unstable with respect to the radial
oscillations but can be stabilized if the conversion timescale between quark
and nucleonic phases at their interface is larger than the oscillation period.
Finally, we review the impact of a transition from high-temperature gapped to
low-temperature gapless two-flavor phase on the thermal evolution of hybrid
stars.
