TL;DR
This paper explores the limitations of the Cardy-Verlinde entropy formula in higher-dimensional FRW universes with general equations of state, showing it does not hold universally when the parameter w differs from 1/n.
Contribution
It generalizes previous results to arbitrary equations of state and demonstrates the breakdown of the entropy formula's applicability beyond specific conditions.
Findings
Entropy cannot be expressed in Cardy-like form for w ≠ 1/n
The entropy formula does not match Friedmann equations when Casimir energy bounds are saturated
Casimir energy bounds do not always lead to Hubble and Bekenstein entropy bounds
Abstract
We generalize the results of hep-th/0008140 to the case of the (n+1)-dimensional closed FRW universe satisfying a general equation of state of the form p=w\rho. We find that the entropy of the universe can no longer be expressed in a form similar to the Cardy formula, when w\neq 1/n. As a result, in general the entropy formula does not coincide with the Friedmann equation when the conjectured bound on the Casimir energy is saturated. Furthermore, the conjectured bound on the Casimir energy generally does not lead to the Hubble and the Bekenstein entropy bounds.
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