  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, $M_{in}$, above the
supersymmetric gauge coupling unification scale, $M_{GUT}$. As in the
constrained MSSM (CMSSM), we assume that the scalar masses and gaugino masses
have common values, $m_0$ and $m_{1/2}$ respectively, at $M_{in}$, as do the
trilinear soft supersymmetry-breaking parameters $A_0$. 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, $m_h$. We find regions of $m_0$, $m_{1/2}$, $A_0$ 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 $m_0$ and $m_{1/2}$ 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 $\tan \beta
\lesssim 6$.
