The cooling dynamics of a 2D granular gas of elongated particles is analyzed. We perform simulations on the temporal evolution of soft particles, using a molecular dynamics algorithm. For weakly dissipative particles, we found a homogeneous cooling process where the overall translational kinetic energy decreases analogously to viscoelastic circular particles. In contrast, for strongly dissipative particles we observed an inhomogeneous cooling process where the diminishing of translational kinetic energy notably slows down. The rotational kinetic energy, however, always decays in agreement with Haff's prediction for the homogeneous cooling state of inelastic particles. We mainly found that the cooling kinetics of the system is controlled by the mechanisms that determine the local energy dissipation (collisions). However, we detected a strong influence of particle shape and inelasticity on the structure of the clusters which develop in the inhomogeneous cooling regimes. Our numerical outcomes suggest that strong dissipation and particle anisotropy induce the formation of ordered cluster structures that retards the relaxation to the final asymptotic regime.