  We report analytically known states at non-zero temperature which may serve
as a powerful tool to reveal common topological and thermodynamic properties of
systems ranging from the QCD phase diagram to topological phase transitions in
condensed matter materials. In the holographically dual gravity theory, these
are analytic solutions to a five-dimensional non-linear-sigma (Skyrme) model
dynamically coupled to Einstein gravity. This theory is shown to be
holographically dual to $\mathcal{N}=4$ Super-Yang-Mills theory coupled to an
$SU(2)$-current. All solutions are fully backreacted asymptotically Anti-de
Sitter~(AdS) black branes or holes. One family of global AdS black hole
solutions contains non-Abelian gauge field configurations with positive integer
Chern numbers and finite energy density. Larger Chern numbers increase the
Hawking-Page transition temperature. In the holographically dual field theory
this indicates a significant effect on the deconfinement phase transition.
Black holes with one Hawking temperature can have distinct Chern numbers,
potentially enabling topological transitions. A second family of analytic
solutions, rotating black branes, is also provided. These rotating solutions
induce states with propagating charge density waves in the dual field theory.
We compute the Hawking temperature, entropy density, angular velocity and free
energy for these black holes/branes. These correspond to thermodynamic data in
the dual field theory. For these states the energy-momentum tensor,
(non-)conserved current, and topological charge are interpreted.
