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We show that the notion of “Hookean law” F=kx, suitably defined in asymptotically flat singly spinning Myers-Perry black hole space-times in dimensions d⩾5, is related to the Emparan-Myers fragmentation (splitting of a black hole into two becomes thermodynamically preferable). Specifically, the values of black hole parameters when fragmentation occurs correspond to the maximal value of F. Furthermore this always happens before F reaches 1/4 in Planck units. These results suggest that a version of “maximum force conjecture” may be relevant for black hole thermodynamics. We also relate these findings to the Ruppeiner thermodynamic geometry of these black holes and speculate on the implications for the underlying microstructures of black hole horizons.