We deal with a generalization of the Principle of Perturbative Agreement (PPA) stated by Hollands and Wald. The PPA requires that, whenever an interacting theory is given by a quadratic perturbation of the free dynamics (so that the whole theory remains "free", i.e. linear equations of motion), the perturbative construction of the algebra of observables and the exact construction should agree, up to a suitable choice of renormalization constants. We develop a proof of the validity of this principle in the case of scalar fields and quadratic interactions without derivatives which differs in strategy from the one given by Hollands & Wald for the case of quadratic interactions encoding a change of metric. Afterwards we generalize the PPA to the situation where a higher order polynomial interaction is present in addition to the exactly tractable quadratic potential. Finally we apply the previous results in order to extend the construction of an interacting KMS state provided by Fredenhagen and Lindner, to the case of the massless Klein Gordon field.