I describe work with Stefan Hollands that establishes a new criterion for the dynamical stability of black holes and black branes with respect to axisymmetric perturbations. Our analysis is done in vacuum general relativity without a cosmological constant in $D \geq 4$ spacetime dimensions, but our approach is applicable to much more general situations. We show that the positivity of the canonical energy, $\mathcal E$, on a subspace of linearized solutions that have vanishing linearized ADM mass and angular momentum implies mode stability. Conversely, failure of positivity of canonical energy on this subspace implies instability in the sense that there exist perturbations that cannot asymptotically approach a stationary perturbation. We further show that the canonical energy is related to the second order variations of mass, angular momentum, and horizon area by $\mathcal E = \delta^2 M - \sum_i \Omega_i \delta^2 J_i - (\kappa/8\pi) \delta^2 A$. This establishes that dynamic stability of a black hole is equivalent to its thermodynamic stability (i.e., its area, $A$, being a maximum at fixed ``state parameters'' $M$, $J_i$). For a black brane, we further show that a sufficient condition for instability is the failure of the Hessian of $A$ with respect to $M$, $J_i$ to be negative, thus proving a conjecture of Gubser and Mitra. We also prove that positivity of $\mathcal E$ is equivalent to the satisfaction of a ``local Penrose inequality,'' thus showing that satisfaction of this local Penrose inequality is necessary and sufficient for dynamical stability.