  We construct an equation of state including the hadron-quark phase
transition. The mixed phase is obtained by the Gibbs conditions for finite
temperature. We adopt the equation of state based on the relativistic mean
field theory for the hadronic phase taking into account pions. As for the quark
phase, the MIT bag model of the deconfined 3-flavor strange quark matter is
used. As a result, our equation of state is thermodynamically stable and
exhibits qualitatively the desired properties of hadron-quark mixed matter,
such as the temperature dependence of the transition density. The pions raise
the transition density because they make the equation of state softer. Using
the equation of state constructed here, we study its astrophysical
implications. The maximum mass of compact stars is investigated, and our
equation of state is consistent with recent observations. We also compute the
collapse of a massive star with 100 solar masses using our equation of state
and find that the interval time from the bounce to the black hole formation
becomes shorter for the model with pions and quarks. The pions and quarks
affect the total energy of the emitted neutrinos because the duration time of
the neutrino emission becomes shorter. The neutrino luminosity rises under the
effect of pions since the density of the proto-neutron star becomes high.
