  The local thermodynamic stability of a black hole (BH) in the canonical
ensemble is defined by the positivity of the specific heat at constant global
charges. Schwarzschild BHs in thermodynamic equilibrium with an energy
reservoir are always unstable against small fluctuations of energy, whereas
sufficiently near-extremal Reissner-Nordstr\"{o}m/Kerr BHs are stable. One
could expect that asymptotically-flat hairy BHs branching off from such stable
phases would also be, by continuity, locally thermodynamically stable for
vanishingly little hair. We show this is not the case in some models, including
scalarized BHs bifurcating from Reissner-Nordstr\"{o}m and spinning BHs with
synchronized hair bifurcating from Kerr. Specifically, it is found that
quasi-bald BHs are locally thermodynamically unstable in the canonical ensemble
for all global charges and regardless of being dynamically and entropically
preferred over bald ones at fixed global charges.
