A particle in the attractive Coulomb field has an interesting property: its specific heat is constant and negative. We show, both analytically and numerically, that when a classical Hamiltonian system stays in weak contact with one such negative specific heat object, its statistics conforms to a fat-tailed power-law distribution with power index given by C/k(B) - 1, where k(B) is the Boltzmann constant and C is the heat capacity. Copyright (C) EPLA, 2012