We revisit minimal supersymmetric SU(5) grand unification (GUT) models in which the soft supersymmetry-breaking parameters of the minimal supersymmetric Standard Model (MSSM) are universal at some input scale, Min, above the supersymmetric gauge-coupling unification scale, MGUT. As in the constrained MSSM (CMSSM), we assume that the scalar masses and gaugino masses have common values, m0 and m1 / 2, respectively, at Min, as do the trilinear soft supersymmetry-breaking parameters A0. Going beyond previous studies of such a super-GUT CMSSM scenario, we explore the constraints imposed by the lower limit on the proton lifetime and the LHC measurement of the Higgs mass, mh. We find regions of m0, m1 / 2, A0 and the parameters of the SU(5) superpotential that are compatible with these and other phenomenological constraints such as the density of cold dark matter, which we assume to be provided by the lightest neutralino. Typically, these allowed regions appear for m0 and m1 / 2 in the multi-TeV region, for suitable values of the unknown SU(5) GUT-scale phases and superpotential couplings, and with the ratio of supersymmetric Higgs vacuum expectation values tan β≲ 6.