A reformer substrate for supporting a catalyst in a hydrocarbon reformer, comprising a graded structure that is inhomogeneous either radially, longitudinally, or both. The inhomogeous graded structure components are selected and arranged to maintain the catalyst operating temperature during extended periods of catalytic inactivity. Selection is based primarily on heat capacity and/or thermal loss properties. Generally, the perimeter of the substrate, radially and/or longitudinally comprises a thick wall of high thermal mass materials to reduce conductive and radiated heat loss, and a high thermal capacity material within the substrate to reduce radiated heat loss. Preferred materials are open-cell rigid foams such as zirconia-toughened alumina reticulated foam, for negative thermal loads in endothermic reaction regimes, and zirconia-mullite honeycomb monolith, for positive or neutral thermal loads in exothermic or autothermic reaction regimes.