In the present paper, we investigate perfect fluid spacetimes and perfect fluid generalized Roberston-Walker spacetimes that contain a torse-forming vector field satisfying almost hyperbolic Ricci solitons. We show that the perfect fluid spacetimes that contain a torse-forming vector field satisfy an almost hyperbolic Ricci soliton, and we prove that a perfect fluid generalized Roberston-Walker spacetime satisfying an almost hyperbolic Ricci soliton $(g,\zeta ,\varrho ,\mu )$ is an Einstein manifold. Also, we study an almost hyperbolic Ricci soliton $(g,V,\varrho ,\mu )$ on these spacetimes when $V$ is a conformal vector field, a torse-forming vector field, or a Ricci bi-conformal vector field.

This work is devoted to the study of Riemann solitons in the setting of the energy-momentum squared gravity framework, expressed as $f(r,T^2)$, regarded as a deformation of Einstein's general relativity. Our attention is placed on the particular model $f(r,T^2)=r+\lambda T^2$, coupled with a perfect fluid, where the dynamics naturally admit Riemann solitons. Within the steady formulation of such solitons, we obtain the corresponding fluid relation of state under $f(r,T^2)$-gravity. Moreover, by exploiting the solitonic structure, we further analyze the admissibility of energy conditions, the emergence of black hole geometries, and the manifestation of singularities in the presence of a perfect fluid for this modified gravitational scenario.