The kinetic response of an L-3 (sponge) phase formed in the C12E5-n-decane-brine system is studied using the Joule-heating temperature jump (JHTJ) technique. The equilibrium state of the spongelike membrane is instantaneously perturbed, and the kinetic response is monitored using a multi-angle light scattering setup. These measurements yield a time-dependent scattering intensity as a function of temperature, scattering vector q, and concentration. We observed a single-exponential relaxation characteristic time, tau(m). The q dependence of the scattering amplitude shows an Omstein-Zernike behavior, but we can identify two concentration regimes with respect to the relaxation behavior. For volume fractions Phi(m) = 0.20, there is no detectable q dependence of the relaxation times, while for samples with Phi(m) > 0.30, the relaxation times display the q(-2) dependence typical of a diffusive process and with relaxation times consistent with those found in dynamic light scattering. At the intermediate concentrations, there is a transition from the q-independent to tile q(-2) dependence behavior. Analysis from each concentration regime reveals distinct differences in the dependence of tau(m) on temperature and concentration, with an extraordinarily strong concentration dependence of tau(m) (tau(m) approximate to Phi(-9)) in the low concentration regime and a temperature dependence corresponding to a formal Arrhenius activation energy of 720 kJ/mol or 275 kT.