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The local thermodynamic stability of a black hole (BH) in the canonicalensemble is defined by the positivity of the specific heat at constant globalcharges. Schwarzschild BHs in thermodynamic equilibrium with an energyreservoir are always unstable against small fluctuations of energy, whereassufficiently near-extremal Reissner-Nordstr\"{o}m/Kerr BHs are stable. Onecould expect that asymptotically-flat hairy BHs branching off from such stablephases would also be, by continuity, locally thermodynamically stable forvanishingly little hair. We show this is not the case in some models, includingscalarized BHs bifurcating from Reissner-Nordstr\"{o}m and spinning BHs withsynchronized hair bifurcating from Kerr. Specifically, it is found thatquasi-bald BHs are locally thermodynamically unstable in the canonical ensemblefor all global charges and regardless of being dynamically and entropicallypreferred over bald ones at fixed global charges.

A little hair can make a big difference: thermodynamic stability of quasi-bald asymptotically-flat black holes