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tokens: stars ( Dutra et al. 2014 ; Lourenco et al. 2019 ). From the pool of models studied in Dutra et al. (2014) ; Lourenco et al. (2019) , we chose four: BSR8, BSR9, G2*, and IU-FSU – all of them allow for the DU process to set in at reasonable neutron star masses (between 1.0 and 2.0 solar masses). Furthermore, all of them present microscopic properties different enough to allow us to conclude that the results we find are most likely general. The same behavior is shared among all models studied, with the only difference being the star’s mass at which the DU process becomes available. This paper is divided as follows: in Sect. 2 we discuss the microscopic models used, Sect. 3 is devoted to the review of the thermal evolution of neutron stars, Sect. 4 contains our results for the relaxation time of neutron stars at the onset of the direct Urca process, and in Sect. 5 we present our conclusions. 2. Microscopic models Quantum hadrodynamics (QHD) is a powerful tool used to build models that represent the strongly interacting matter with hadrons being the main degrees of freedom. The first model constructed from such an approach was proposed by Walecka (1974) and Serot Walecka (1997) with the two free parameters fixed to reproduce the nuclear matter energy per particle as a function of the density, and a minimum of B 0 = −15.75 MeV at the saturation density ρ 0 = 0.19 fm −3 . However, the model also presents bad results for the effective mass ratio and incompressibilty, both at ρ = ρ 0 , namely, ( M n is the nucleon rest mass) and K 0 = 540 MeV, respectively. Over the years, many other improved versions of this model were proposed in which and K 0 are fixed to more compatible values with experimental/theoretical predictions. Furthermore, other bulk parameters at the saturation density are also used in order to constrain the free coupling constants of these microscopic models. Here, we investigate the cooling process of neutron stars based on parametrizations of a general model described by the following Lagrangian density ( Dutra et al. 2014 ; Li et al. 2008 ) in which F μ ν = ∂ μ ω ν − ∂ ν ω μ and B μ ν = ∂ μ ρ ν − ∂ ν ρ μ . ψ is the nucleon field and σ , ω μ , and ρ μ represent the fields of the mesons σ , ω , and ρ , respectively. The mean-field approximation is used in order to solve the equations of motion for the fields. This procedure, along with the energy-momentum tensor, T μ ν , allows the construction of all thermodynamics of the system since the energy density and pressure are given by E = ⟨ T 00 ⟩ and P = ⟨ T i i ⟩/3, respectively. These equations of state are evaluated, as a function of the density, by taking into account the auto-consistency of the field equations and the definition of the effective nucleon mass given by M * = M n − g σ σ . More details related to the calculations of these quantities can be found in Dutra et al. (2014) and Li et al. (2008) , and references therein. In order to study neutron stars and their thermal