  We introduce an ingenious approach to explore cosmological implications of
higher-derivative gravity theories. The key novelty lies in the
characterization of the additional massive spin-0 modes constructed from Hubble
derivatives as an effective density, with the corresponding pressure uniquely
determined by energy conservation, while terms with no Hubble derivatives
directly alter Friedmann equations. This classification of the various
high-derivative contributions to Friedmann equations develops insight about
their cosmological impacts and is essential for understanding the universe's
evolution across energy scales. Various examples of higher-derivative gravity
theories illustrate the power of this method in efficiently solving Friedmann
equations and exploring new phenomena. Using CMB and BAO data, we apply this
method to assess the observational feasibility of wall-bouncing universes, as
predicted by scenarios with, e.g., certain third order modifications to general
relativity. These models also provide an inflationary phase without the need to
introduce extra scalar fields.
