The CUORE experiment is a large bolometric array searching for the lepton number violating neutrino-less double beta decay (0νββ) in the isotope ¹³⁰Te. In this work we present the latest results on two searches for the double beta decay (DBD) of ¹³⁰Te to the first 0⁺₂ excited state of ¹³⁰Xe: the 0νββ decay and the Standard Model-allowed two-neutrinos double beta decay (2νββ). Both searches are based on a 372.5 kg×yr TeO₂ 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 S^0ν_1/2 = 5.6×10²⁴ yr for the 0νββ decay and S^2ν_1/2 = 2.1×10²⁴ yr for the 2νββ 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: (T_1/2)^0ν_0⁺2 > 5.9×10²⁴ yr for the 0νββ mode and (T_1/2)^2ν_0⁺2 > 1.9×10²⁴ yr for the 2νββ mode. These represent the most stringent limits on the DBD of ¹³⁰Te to excited states and improve by a factor ~5 the previous results on this process.