  We study the observability for the flavor-changing decay of a top quark $t\to
ch$ at the LHC and future hadron colliders, namely, High-Luminosity LHC
(HL-LHC), High-Energy LHC (HE-LHC) and Future Circular hadron-hadron Collider
(FCC-hh). Two scenarios in which the Higgs boson could decay: into a quark
bottom pair $(\textit{bb-channel})$ and two photons
$(\textit{$\gamma\gamma$-channel})$ are analyzed. A Monte Carlo analysis of the
signal and the Standard Model (SM) background is computed. Center-of-mass
energies of $\sqrt{s}=14,\;27\; $\text{and}$\; 100$ TeV and integrated
luminosities from $0.3$ to $30$ ab$^{-1}$ are explored. The theoretical
framework adopted in this work is the Type-III Two-Higgs Doublet Model
(THDM-III) for which, constraints on the parameter space from the Higgs boson
coupling modifiers $\kappa_i$ are presented and used in order to evaluate the
branching ratio of the $t\to ch$ decay and the ($pp\to t\bar{t},\,t\to ch$)
production cross section. We find that with the integrated luminosity achieved
at the LHC, the $t\to ch$ decay is out of the reach of detection. More
promising results emerge for the HL-LHC, HE-LHC and FCC-hh in which potential
discoveries could be claimed.
