  $f(P)$ gravity is a novel extension of ECG in which the Ricci scalar in the
action is replaced by a function of the curvature invariant $P$ which
represents the contractions of the Riemann tensor at the cubic order \cite{p}.
The present work is concentrated on bounding some $f(P)$ gravity models using
the concept of energy conditions where the functional forms of $f(P)$ are
represented as \textbf{a)} $f(P) = \alpha \sqrt{P}$, and \textbf{b)} $f(P) =
\alpha \exp (P)$, where $\alpha$ is the sole model parameter. Energy conditions
are interesting linear relationships between pressure and density and have been
extensively employed to derive interesting results in Einstein's gravity, and
are also an excellent tool to impose constraints on any cosmological model. To
place the bounds, we ensured that the energy density must remain positive, the
pressure must remain negative, and the EoS parameter must attain a value close
to $-1$ to make sure that the bounds respect the accelerated expansion of the
Universe and are also in harmony with the latest observational data. We report
that for both the models, suitable parameter spaces exist which satisfy the
aforementioned conditions and therefore posit the $f(P)$ theory of gravity to
be a promising modified theory of gravitation.
