  We discuss the thermal leptogenesis mechanism within the minimal gauged
U(1)$_{L_\mu-L_\tau}$ model to explain the observed baryon asymmetry of the
Universe (BAU). In such framework, the phases of the
Pontecorvo-Maki-Nakagawa-Sakata neutrino mixing matrix and the sum of the
Standard Model neutrino masses are predictable because of a restricted neutrino
mass matrix structure. Additionally, in the context of thermal leptogenesis,
the BAU can be computed in terms of the three remaining free variables that
parameterise the right-handed neutrino masses and their Yukawa couplings to the
Higgs and lepton doublets. We identify the ranges of such parameters for which
the correct BAU can be reproduced. We adopt the formalism of the density matrix
equations to fully account for flavour effects and consider the decays of all
the three right-handed neutrinos. Our analysis reveals that thermal
leptogenesis is feasible within a wide parameter space, specifically for Yukawa
couplings ranging from approximate unity to $\mathcal{O}(0.03-0.05)$ and mass
of the lightest right-handed neutrino $M_1\gtrsim 10^{11-12}\,\text{GeV}$,
setting a leptogenesis scale in the considered model which is higher than that
of the non-thermal scenario.
