We investigate which general results concerning the local stability of steady states of arbitrary chemical reaction networks can be deduced with the Glansdorff-Prigogine stability criterion. Especially, it is proven that the presence of an autocatalytic reaction is not a necessary condition for a violation of the thermodynamic stability condition. It turns out that every reaction with at least one variable reactant at each side of the reaction equation can potentially destabilize the steady states. An explicit example of a simple reaction system without autocatalytic reactions where the stability of the steady state changes via a supercritical Hopf bifurcation is discussed. Furthermore, in expanding the original concept for proving local stability to global stability analyses, a general way for constructing different Lyapunov functions is given.