Catalyst degradation is one major challenge preventing the worldwide commercialization of the Proton Exchange Membrane Fuel Cells. In this study, we investigate the development of a novel hierarchical carbonaceous support for the platinum catalysts, called graphene-carbon nanotube hybrids (GCNT), and its degradation behavior during an accelerated degradation test. The carbon support is fabricated by growing graphene directly onto carbon nanotubes to form a unique all-carbon nanostructure possessing both an ultra-high density of exposed graphitic edges of graphene and a porous structure of carbon nanotubes. The GCNT-supported platinum catalyst exhibits a higher intrinsic catalytic activity than a carbon black-supported platinum catalyst, and much higher than a CNT-supported platinum catalyst. The enhanced catalytic activity of the GCNT-supported platinum catalyst is explained by the high graphitic edge density which promotes the catalytic reactions on platinum catalyst. The GCNT-supported platinum catalyst also exhibits a superior electrochemical stability over that of the carbon black-supported platinum catalyst, explained by the high crystallinity of the GCNT support. The superior stability is expressed by a lower loss in polarization performance, a smaller increase in charge transfer resistance, a lower loss in the platinum electrochemical surface area, a lower rate of carbon corrosion, and a more stable catalyst microstructure. Energy security and climate change have become major global concerns, and viable candidates for renewable energy technologies are actively being sought. With high energy conversion efficiencies and low emissions, fuel cell technologies have received much research attention recently.