observed that GSLT is verified at apparent horizon
but not at event horizon for a specific model of dark energy.
where [M.sub.ADM] is the ADM mass and [A.sup.+.sub.min] ([A.sup.-.sub.min]) is the minimum area required to enclose the outermost future (past) apparent horizon
. The Riemannian version of this inequality in the time-symmetric case, when k = 0, along with the corresponding rigidity statement in the case of equality, has been established.
In the present work, we develop the particle creation scenario in the dynamical Chern-Simons modified gravity by assuming various modified entropies (Bekenstein entropy logarithmic entropy, power law correction, and Renyi entropy) on the apparent horizon
. In the presence of particle creation rate, we discuss the generalized second law of thermodynamics (GSLT).
Then, one can derive Hawking temperature on the apparent horizon
Sadjadi  studied the second law as well as GSLT in f(R, G) gravity for de Sitter universe model as well as power-law solution with the assumption that apparent horizon
is in thermal equilibrium.
In this paper we want to connect the entanglement energy of quantum particles across the apparent horizon
, which is missed in the classical cosmological equations written for the Hubble volume, with the dark energy (DE), origin of which remains a mystery .
For a spatially flat FRW universe with Hubble constant H and apparent horizon
[r.sub.A] = 1/H, we have
In [1,5], Akbar and Cai reversed the formulation by rewriting the Friedmann equations into the heat balance equation and the unified first law of thermodynamics at the cosmical apparent horizon
, for General Relativity (GR), Gauss-Bonnet, and Lovelock gravity.
From the viewpoint of thermodynamics, if OCL is valid, then by applying the thermodynamics laws to the apparent horizon
, as the proper causal boundary located at [2-4]
According to [47, 48], it is suitable to use the notion [??] = f(E[l.sub.p])a(t)r/g(E[l.sub.p]) and relation [h.sup.[mu]v][[partial derivative].sub.[mu]][??][[partial derivative].sub.v][??] = 0; the modified dynamical apparent horizon
of FRW space-time becomes
Dehghani, the authors tackled the unified first law of thermodynamics and its genuine connection with the apparent horizon
of FRW universe in which they found that whenever there is no energy exchange between the various parts of cosmos, one could get an expression for the apparent horizon
entropy in quasi-topological gravity.
In the present work, we are going to study the thermodynamics aspects of this nonminimally coupled f(T) model at the apparent horizon
of an expanding cosmological background.
Recently the quantum tunnelling of scalar and Dirac particles from the apparent horizon
in a FRW universe has been investigated in [29-32].
In order to achieve this goal, applying the thermodynamics laws to the apparent horizon
of FRW universe, we find an expression for the horizon entropy in Section 2.
This "information paradox" recently led to the suggestion that black holes only possess apparent horizons
 as opposed to genuine event horizons: features traditionally regarded as the defining hallmarks of true black holes .