  In this paper, we consider projective deformation of the geodesic system of
Finsler spaces by holonomy invariant functions: Starting by a Finsler spray $S$
and a holonomy invariant function $P$, we investigate the metrizability
property of the projective deformation $\widetilde{S}=S-2\lambda P C$. We prove
that for any holonomy invariant nontrivial function $P$ and for almost every
value $\lambda\in R$, such deformation is not Finsler metrizable. We identify
the cases where such deformation can lead to a metrizable spray: in these
cases, the holonomy invariant function is necessarily one of the principal
curvatures of the geodesic structure.
