We consider a system of Brownian particles confined in a two-dimensional bilobal enclosure whose walls are driven in time periodically by an external perturbation. The response of the particles under shape modulation is characterized by a relaxational delay which results in a non-vanishing area of the response function—field loop, response function being the integrated probability of residence of the particles in any of the lobes. This phenomenon is an entropic analogue of dynamical hysteresis, which vanishes in the quasi-static limit. The hysteresis loop area depends on temperature, strength of modulating field, and the geometrical parameters of the enclosure and exhibits a turnover as a function of frequency of the field.