The CUORE experiment is a large bolometric array searching for the lepton number violating neutrino-less double beta decay ($0 u\beta\beta$) 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 $0 u\beta\beta$ decay and the Standard Model-allowed two-neutrinos double beta decay ($2 u\beta\beta$). 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 ${0 u\beta\beta}$ decay and $\mathrm{S^{2 u}_{1/2} = 2.1 \times 10^{24} \: \mathrm{yr}}$ for the ${2 u\beta\beta}$ 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 $0 u\beta\beta$ mode and $\mathrm{(T_{1/2})^{2 u}_{0^+_2} > 1.3 \times 10^{24} \: \mathrm{yr}}$ for the $2 u\beta\beta$ 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.