  We investigate the thermodynamics of the anisotropic magnetic $\rm AdS_5$
black brane solution found by D'Hoker and Kraus arXiv:0908.3875. This solution
is the gravity dual of a strongly coupled ${\cal N}=4$ Super Yang-Mills plasma
in ${\mathbb R}^{3,1}$, with temperature $T$, in the presence of a magnetic
field ${\cal B}$. Following the procedure of holographic renormalisation we
calculate the Gibbs free energy and the holographic stress tensor of the
conformal plasma. We evaluate several thermodynamic quantities including the
magnetisation, the anisotropic pressures and the speeds of sound. Our results
are consistent with an RG flow from a perturbed $\rm AdS_5$ black brane at
small ${\cal B}/T^2$ to a $\rm \bf BTZ \times {\mathbb R}^2$ black brane at
large ${\cal B}/T^2$. We also perform a phenomenological analysis where we
compare the thermodynamics of a magnetised conformal plasma against the lattice
QCD results for the thermodynamics of the magnetised quark-gluon plasma.
