In a recent paper (Santillán et al 2001 J. Phys. D: Appl. Phys. 34 2068-72) the local stability of a Curzon-Ahlbom-Novikov (CAN) engine with equal conductances in the coupling with thermal baths was analysed. In this work, we present a local stability analysis of an endoreversible engine operating at maximum power output, for common heat transfer laws, and for different heat conductances α and β, in the isothermal couplings of the working substance with the thermal sources T1 and T2 (T 1 > T2). We find that the relaxation times, in the cases analysed here, are a function of α, β, the heat capacity C, T1 and T2. Besides, the eigendirections in a phase portrait are also functions of τ = T1/T2 and the ratio β/α. From these findings, phase portraits for the trajectories after a small perturbation over the steady-state values of internal temperatures are presented, for some significant situations. Finally, we discuss the local stability and energetic properties of the endoreversible CAN heat engine. © 2005 IOP Publishing Ltd.