Generalized Misner-Sharp energy in $f(R,\mathcal{G})$ gravity

In this work, we explore the formulation of the Misner-Sharp energy within the framework of $f(R, \mathcal{G})$ gravity, a modified theory incorporating the Ricci scalar $R$ and the Gauss-Bonnet scalar $\mathcal{G}$. By extending the quasilocal energy definition to both static spherically symmetric spacetime and the dynamic Friedmann-Lemaitre-Robertson-Walker (FLRW) spacetime, we derive explicit expressions for the generalized Misner-Sharp energy using two complementary approaches: the integration method and the conserved charge method based on the Kodama vector. Our analysis shows that the Misner-Sharp energy expression in $f(R, \mathcal{G})$ gravity reduces to standard $f(R)$ gravity results when the Gauss-Bonnet term is absent, revealing how curvature modifications influence the geometric structure and dynamics of cosmic evolution. Furthermore, we investigate the thermodynamic properties at the apparent horizon associated with the FLRW background, and we find a connection to non-equilibrium thermodynamics unique to $f(R, \mathcal{G})$ gravity. These findings underscore the subtle and fundamental role of curvature corrections in determining the energy distribution and thermodynamic behavior of gravitational systems.