In the Brandenberger-Vafa scenario of string gas cosmology, the Universe starts as a small torus of string length dimension filled with a hot gas of strings. In such extreme conditions, in addition to the departure from Einstein gravity which is due to the dilaton, one expects higher curvature corrections to be relevant. Motivated by this fact, we study the effect of the leading α′3 corrections in type IIB string theory for this scenario. Within the assumptions of: weak coupling, adiabatic evolution and thermodynamical equilibrium, we perturbatively solved the corresponding equations of motion in two different cases: (i) the isotropic case which is governed by a single scale factor and (ii) the anisotropic case given by two different scale factors. In the first case, we consider two regimes: (ia) the Hagedorn regime where the string gas equation of state is that of pressureless dust and (ib) the radiation regime. In the second case, (ii), we only considered a radiation-like equation of state. We found that the inclusion of α′ corrections affects the scale factor(s) in the opposite way in the Hagedorn and in the radiation regimes, acting as a driving force for the first one and a damping force for the second one. This effect is small for reasonable initial conditions and it is only observed at early times. Moreover it is bigger in the Hagedorn regime than in the radiation regime. We also analysed the fixed points of the system. We found that there exists a stable dS fixed point, which does not exist when the corrections are neglected.