  In this work we present the generalization of some thermodynamic properties
of the black body radiation (BBR) towards an $n-$dimensional Euclidean space.
For this case the Planck function and the Stefan-Boltzmann law have already
been given by Landsberg and de Vos and some adjustments by Menon and Agrawal.
However, since then no much more has been done on this subject and we believe
there are some relevant aspects yet to explore. In addition to the results
previously found we calculate the thermodynamic potentials, the efficiency of
the Carnot engine, the law for adiabatic processes and the heat capacity at
constant volume. There is a region at which an interesting behavior of the
thermodynamic potentials arise, maxima and minima appear for the $n-d$ BBR
system at very high temperatures and low dimensionality, suggesting a possible
application to cosmology. Finally we propose that an optimality criterion in a
thermodynamic framework could have to do with the $3-d$ nature of the universe.
