Modified Cosmological Curvature and Its Implications on Spacetime Metrics

We explore a novel formulation of cosmological curvature given by:\begin{equation} R_{\text{new}} \;=\; \frac{R_{\text{standard}}}{1 \;+\; \gamma \,R_{\text{standard}}}\;, \label{eq:new_curvature_main}\end{equation}where \(\gamma\) is a deformation parameter indicating small deviations from standard General Relativity. We derive and analyze the corresponding modified metric and present extensive mathematical derivations demonstrating how these deformations can yield new dynamical features in both early- and late-time cosmology. Building on horizons-entropy parallels from Ref.~\cite{kruglov2025}, we highlight how curvature modifications and generalized thermodynamic considerations can intertwine, suggesting possible novel contributions to the effective energy budget of the universe and to the onset of cosmic acceleration. Numerical studies and analytical solutions illustrate the modified Hubble evolution, phase transitions between decelerating and accelerating epochs, and the evolution of dark-energy-like components emerging from the new geometry.