I. INTRODUCTION Neutron stars are natural laboratories for studying the physics of dense matter. They are associated with some of the most exotic environments in the universe. Our knowledge of neutron star interiors is still uncertain. The central density of neutron stars can be extremely high, and many possibilities for such dense matter have been suggested [1–3]. For densities below twice normal nuclear matter density ( ρ$_{0}$ ∼ 0 . 15 fm − $^{3}$), the matter consists of only nucleons and leptons. When the density is higher than 2 ρ$_{0}$ , the equation of state (EOS) and composition of matter are much less certain. The presence of hyperons in neutron stars has been studied by many authors [4–8]. K − condensation in dense matter was suggested by Kaplan and Nelson [9] and has been extensively discussed in many works [2, 4, 10]. It has been suggested that the quark matter may exist in the core of massive neutron stars, and the hadron-quark phase transition can proceed through a mixed phase of hadronic and quark matter [1, 2, 6, 11]. If deconfined quark matter does exist inside stars, it is likely to be in a color superconducting phase [1, 6], and various color superconducting phases have been intensively investigated in recent years [12–14]. Baryon pairing is believed to play an important role in the evolution of neutron stars [2, 15, 16]. The presence of neutron superfluidity in the crust and the inner part of neutron stars can be considered well established [15]. The neutron fluid in the crust probably forms a $^{1}$S$_{0}$ superfluid. With increasing density, the $^{1}$S$_{0}$ interaction turns repulsive, and the neutrons in the outer core mainly form a $^{3}$P$_{2}$ superfluid. On the other hand, one can expect $^{1}$S$_{0}$ proton pairing in the outer core because the small proton fraction brings about a low proton density in this region. In the inner core of neutron stars, hyperons may appear through the weak interaction because of the fast rise of the baryon chemical potentials with density. It is widely accepted that hyperons appear around 2 ρ$_{0}$ . The presence of hyperons tends to soften the EOS at high density and lower the maximum mass of neutron stars [4–8] as well as increase the neutron star cooling rate [16, 17]. In general, the first hyperon to appear is Λ, which is the lightest one with an attractive potential in nuclear matter. The potential of Σ hyperons is now considered to be repulsive; therefore Σ − appears at a higher density than Λ in neutron star matter [18, 19]. The $^{1}$S$_{0}$ superfluidity of Λ hyperons is suggested to occur in the same way as that of neutrons arising from the attractive ΛΛ interaction in the $^{1}$S$_{0}$ channel [20–24]. It is known that hyperon pairing can significantly affect the thermal 2