  We present the first examples of phase transitions in asymptotically flat
black hole solutions. We analyze the thermodynamic properties of black holes in
order $N\ge 3$ Lovelock gravity, with zero cosmological constant. We find a new
type of "inverted" swallowtail indicative of stable temperature regions for an
otherwise unstable neutral black hole, and demonstrate multiple such stable
phases can exist and coexist at multi-critical points. We also find that for
charged black holes, ordinary swallowtails can exist on the stable Gibbs free
energy branch, allowing for multiple first order phase transitions as seen for
AdS black holes. A triple point for $N=5$ and a quadruple point for $N=7$ are
presented explicitly. We investigate changes in the Gibbs free energy as the
lowest order Lovelock constant is varied, and draw comparisons to pressure
changes for AdS black hole systems.
