Vortex cores in under- and overdoped Bi2Sr2CaCu2O81d are studied by local probe tunneling spectroscopy. At the center of the cores, we find a gaplike structure at the Fermi level which scales with the superconducting gap, but no quasiparticle bound states. This low temperature pseudogap is intimately related to the superconducting gap and shows striking similarities with the normal state pseudogap measured above Tc. A possible interpretation is that both pseudogap structures reflect the same “normal” state containing phase incoherent excited pair states.