Is the (3 + 1)-d nature of the universe a thermodynamic necessity?

It is well established that at early times, long before the time of radiation-matter density equality, the universe could have been well described by a spatially flat, radiation only model. In this article we consider the whole primeval universe, as a first approach, as a black-body radiation system in an n-dimensional Euclidean space. We propose that the (3 + 1)-dimensional nature of the universe could be the result of a thermodynamic selection principle stemming from the second law of thermodynamics. In regard to the three spatial dimensions we suggest that they were chosen by means of the minimization of the Helmholtz free energy per hypervolume unit following possibly a kind of broken symmetry process, while the time dimension, as is well known, is related with the principle of increment of entropy for closed systems: the so-called arrow of time.