  The CUORE experiment is a large bolometric array searching for the lepton
number violating neutrino-less double beta decay ($0\nu\beta\beta$) 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 $0\nu\beta\beta$ decay and
the Standard Model-allowed two-neutrinos double beta decay ($2\nu\beta\beta$).
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 $\mathrm{S^{0\nu}_{1/2} = 5.6 \times
10^{24} \: \mathrm{yr}}$ for the ${0\nu\beta\beta}$ decay and
$\mathrm{S^{2\nu}_{1/2} = 2.1 \times 10^{24} \: \mathrm{yr}}$ for the
${2\nu\beta\beta}$ 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: $\mathrm{(T_{1/2})^{0\nu}_{0^+_2} > 5.9 \times 10^{24} \:
\mathrm{yr}}$ for the $0\nu\beta\beta$ mode and
$\mathrm{(T_{1/2})^{2\nu}_{0^+_2} > 1.3 \times 10^{24} \: \mathrm{yr}}$ for the
$2\nu\beta\beta$ 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.
