In a recent numerical investigation of the effect of trapped 'cosmions' on horizontal-branch (HB) stars, thermal pulses were observed to occur if the energy transfer provided by these weakly interacting particles was chosen to be efficient enough to break convection. A simple analytic two-zone model of the core of an HB star is presented which allows the interpretation of these pulsations as oscillations of thermal relaxation between the small energy generating 'nuclear core' and the overlying 'gravothermal buffer'. It is shown that any novel form of energy transfer which breaks convection will lead to this behavior where the usual stellar 'conductive equilibrium' is replaced by a limit cycle. The star follows this cycle, oscillating with a period roughly given by the Kelvin-Helmholtz time scale of the core.