Many theoretical models of physics beyond the Standard Model provide a signature of two photons and large missing transverse energy [special characters omitted] in the final state. This search was performed using 36 pb^–1 of proton-proton collision data at [special characters omitted] = 7 TeV. The candidate events are determined by comparing the [special characters omitted] distribution for events with two photons and at least one hadronic jet with the [special characters omitted] T distribution from QCD and electroweak processes that have no true missing transverse energy. 1.2 ± 0.8 background events were expected, with one event observed. This result was interpreted in two theoretical models beyond the Standard Model. The cross section for General Gauge Mediated Supersymmetry Breaking is given an upper limit between 0.3 pb and 1.1 pb. Universal Extra Dimensions has been excluded for all values of 1/R < 889 GeV. All limits are given at 95% CL.