The calculation of the thermal stratification in the superadiabatic layers of stellar models with convective envelopes is a long-standing problem of stellar astrophysics, and has a major impact on predicted observational properties such as radius and effective temperature. The mixing length theory, almost universally used to model the superadiabatic convective layers, contains one free parameter to be calibrated (αml
) whose value controls the resulting effective temperature. Here we present the first self-consistent stellar evolution models calculated by employing the atmospheric temperature stratification, Rosseland opacities, and calibrated variable αml
(dependent on effective temperature and surface gravity) from a recently published large suite of three-dimensional radiation hydrodynamics simulations of stellar convective envelopes and atmospheres for solar stellar composition. From our calculations (with the same composition of the radiation hydrodynamics simulations), we find that the effective temperatures of models with the hydro-calibrated variable αml
(that ranges between ∼1.6 and ∼2.0 in the parameter space covered by the simulations) present only minor differences, by at most ∼30-50K, compared to models calculated at constant solar αml
(equal to 1.76, as obtained from the same simulations). The depth of the convective regions is essentially the same in both cases. We also analyzed the role played by the hydro-calibrated T(τ) relationships in determining the evolution of the model effective temperatures, when compared to alternative T(τ) relationships often used in stellar model computations. The choice of the T(τ) can have a larger impact than the use of a variable αml
compared to a constant solar value. We found that the solar semi-empirical T(τ) by Vernazza et al. (1981ApJS...45..635V
) provides stellar model effective temperatures that agree quite well with the results with the hydro-calibrated relationships.