The CUORE experiment is a large bolometric array searching for the lepton number violating neutrino-less double beta decay ($0 ubetabeta$) in the isotope $mathrm{^{130}Te}$. In this work we present the latest results on two searches for the double beta decay (DBD) of $mathrm{^{130}Te}$ to the first $0^{+}_2$ excited state of $mathrm{^{130}Xe}$: the $0 ubetabeta$ decay and the Standard Model-allowed two-neutrinos double beta decay ($2 ubetabeta$). Both searches are based on a 372.5 kg$times$yr TeO$_2$ exposure. The de-excitation gamma rays emitted by the excited Xe nucleus in the final state yield a unique signature, which can be searched for with low background by studying coincident events in two or more bolometers. The closely packed arrangement of the CUORE crystals constitutes a significant advantage in this regard. The median limit setting sensitivities at 90% Credible Interval (C.I.) of the given searches were estimated as $mathrm{S^{0 u}_{1/2} = 5.6 times 10^{24} : mathrm{yr}}$ for the ${0 ubetabeta}$ decay and $mathrm{S^{2 u}_{1/2} = 2.1 times 10^{24} : mathrm{yr}}$ for the ${2 ubetabeta}$ decay. No significant evidence for either of the decay modes was observed and a Bayesian lower bound at $90%$ C.I. on the decay half lives is obtained as: $mathrm{(T_{1/2})^{0 u}_{0^+_2} > 5.9 times 10^{24} : mathrm{yr}}$ for the $0 ubetabeta$ mode and $mathrm{(T_{1/2})^{2 u}_{0^+_2} > 1.3 times 10^{24} : mathrm{yr}}$ for the $2 ubetabeta$ mode. These represent the most stringent limits on the DBD of $^{130}$Te to excited states and improve by a factor $sim5$ the previous results on this process.
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