In this work we use momentum-space techniques to evaluate the propagator $G(x,x^{prime})$ for a spin $1/2$ mass dimension one spinor field on a curved Friedmann-Robertson-Walker spacetime. As a consequence, we built the one-loop correction to the effective lagrangian in the coincidence limit. Going further we compute the effective lagrangian in the finite temperature regime. We arrive at interesting cosmological consequences, as time-dependent cosmological `constant, fully explaining the functional form of previous cosmological models.