We show that for critically non-recurrent rational functions all the definitions of topological pressure proposed in  coincide for all t ≥ 0. Then we study in detail the Gibbs states corresponding to the potentials -tlog|f'| and their σ-finite invariant versions. In particular we provide a sufficient condition for their finiteness. We determine the escape rates of critically non-recurrent rational functions. In the presence of parabolic points we also establish a polynomial rate of appropriately modified escape. This extends the corresponding result from  proven in the context of parabolic rational functions. In the last part of the paper we introduce the class of critically tame generalized polynomial-like mappings. We show that if f is a critically tame and critically non-recurrent generalized polynomial-like mapping and g is a Hölder continuous potential (with sufficiently large exponent if f has parabolic points) and the topological pressure satisfies P(g) > sup(g), then for sufficiently small δ >0, the function t↦ P(tg), t ∈ (1-δ,1+δ), is real-analytic.