Thermodynamics of Apparent Horizons in a (2+1)-Dimensional Modified Friedmann-Robertson-Walker Universe

In this paper, thermodynamic effects in modified (2+1)-dimensional FRW cosmology is explored. The impacts of fluctuations in the Hubble parameter and its derivatives on the universe's evolution and corresponding thermodynamic behavior are examined. Using the modified FRW metric as a starting point, and the conditions ∇μr∇μ=0, we obtained the adjusted apparent horizon radius rA-2 = H2 + k/a2 and matching surface gravity. The energy and generalized entropy at the horizon are obtained using the Misner–Sharp formalism. This is accomplished by using our modified (2+1)-dimensional FRW cosmology in conjunction with the unified first law of thermodynamics. The evolution equations were formulated for Ḣ and H2 based on modified Friedmann and acceleration equations with dimensionless constants α1, α2, β1 and β2. The dynamics of (2+1)-dimensional cosmology is improved by these changes. In order to guarantee conformity with the generalized first law, we apply requirements to the modified gravity parameters by examining the consistency of the thermodynamics equation Tds = dE +Wdv. These restrictions result in invariant relations between the corrections coefficients, namely α2/α1 = 1-β2/1-β1 and β2/β1 = 1-α2/1-α1, which are symmetric under the interchange αi ↔βi. This formulation establishes a connection between thermodynamics and gravitational dynamics in (2+1)-dimensions, thereby facilitating the systematic examination of the influence of modified gravity on cosmic evolution.