  Generically, the black brane solution with planar horizons is
thermodynamically stable. We find a counter-example to this statement by
demonstrating that an anisotropic black brane is unstable. We present a charged
black brane solution dual to a spatially anisotropic finite temperature
$\mathcal{N}=4$ super Yang-Mills plasma at finite $U(1)$ chemical potential.
This static and regular solution is obtained both numerically and analytically.
We uncover rich thermodynamic phase structures for this system by considering
the cases when the anisotropy constant "a" takes real and imaginary values,
respectively. In the case $a^2>0$, the phase structure of this anisotropic
black brane is similar to that of Schwarzschild-AdS black hole with $S^3$
horizon topology, yielding a thermodynamical instability at smaller horizon
radii. For the condition $a^2\leq 0$, the thermodynamics is dominated by the
black brane phase for all temperatures.
