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This thesis work focuses on studying the possible existence of phase transitions in the immediate compact remnants of core collapse supernova, neutron stars, and the theoretical models that describe the interior of dense matter. Specifically, we are interested in analyzing the feasibility of a transition from hadronic matter to quark matter in the cores of these objects. The density of matter inside neutron stars is several times that of atomic nuclei, and the equation of state that describes such matter in such a regime is still unknown. In this context, it is known that the interaction between the constituents of nucleons, the quarks, weakens with increasing density due to the intrinsic property of the QCD known as it asymptotic freedom. Therefore, matter should either dissolve into a quark-free state at high densities, or else form a superconduct- ing state of color. This superconducting phase of color would be energetically favorable, if it were present in a cold neutron star, since a system of fermions that interact weakly at low temperature is unstable with respect to the formation of Cooper pairs. Although it is impossible to know both theoretically and experimentally whether these phases exist in neutron stars, the interpolation of the resolvable part of QCD at high densities, together with the hadronic equations of state at low densities, suggest that they could appear in the interior of compact objects. For the phase transition we will use two different formalisms: the Maxwell formalism, in which an abrupt phase transition between hadronic and quark matter without mixed phase formation is assumed, and the Gibbs formalism, in which a mixed phase in which hadrons and quarks coexist. For the description of hadronic matter, we will use different parametrizations of the relativistic mean field model with density-dependent coupling constants. For the description of quark matter we will use an effective nonlocal Nambu Jona-Lasinio model of three flavors with vector interactions, in which we will include the possibility of formation of diquarks to model a superconducting phase of color in SU (3), which we will call 2SC + s. Phase diagrams and equations of state of quark matter at finite temperature are presented, and the influence of that kind of matter on observables associated with neutron stars is investigated. Likewise, using hybrid equations of state, the simplified thermal evolution of compact stars during their formation is studied, from their state of proto-neutron stars to that of cold neutron stars, and the results obtained are compared with recent astrophysical observations. The pa- rameterizations used in this work are adjusted to the most recent measurements of masses and coupling constants of the QCD, which imposes strong restrictions on the existence of quark matter in proto-stars, unlike what happens with less realistic models or with more free parameters. However, the results obtained indicate that even considering these restrictions, the occurrence of quark matter in the nuclei of these stars remains a promis- ing possibility. The remaining free parameters of the models were adjusted taking into account the observational restrictions, coming from precise determinations of the pulsars masses of ∼ 2 M⊙, and the event corresponding to the fusion of two neutron stars, known as GW170817. The fact that the use of more realistic models for the description of the dense matter in these objects indicates the presence of quark matter inside neutron stars, could be an answer to the question of the behavior of that kind of matter and the determination of its corresponding equation of state.