The entropy on

apparent horizon can be defined as follows:

Lastly observe that since [phi] vanishes at [partial derivative][M.sub.0], we must have that [partial derivative][M.sub.0] is an

apparent horizon. However [partial derivative]M is an outermost

apparent horizon, so in fact [partial derivative][M.sub.0] = [partial derivative]M, and hence [M.sub.0] = M.

(iii) Power law correction: in thermodynamics of

apparent horizon in the standard FRW cosmology, the geometric entropy is assumed to be proportional to its horizon area ([S.sub.A] = A/4).

Cao, "Unified first law and the thermodynamics of the

apparent horizon in the FRW universe," Physical Review D, vol.

To study this law, we first find the dynamical

apparent horizon evaluated by the relation

Then the entanglement energy of quantum particles across the

apparent horizon [R.sub.H], which is defined as disturbed vacuum energy due to the presence of a boundary [49], is missed in the cosmological equations written for the Hubble volume and can be taken into account by introduction of a boundary term.

Considering the Hawking temperature (similar to (12), we ignore the direct correction to the radius in the Hawking temperature, and the changes of numbers of degrees of freedom directly stem from the corrections of the area of the

apparent horizon) T = 1/2[pi][r.sub.A] and E = -([rho] + 3p)[??] with dark energy in the bulk, we obtain

Dynamics of the Cosmical

Apparent Horizon in Eddington-Born-Infeld Gravity

These equations would also be obtainable by modifying the Einstein field equations as [G.sub.[mu][nu]] = 8[pi][T.sub.[mu][nu]](e), in agreement with attributing the Bekenstein entropy to the

apparent horizon (see the paragraph after (16)).

In the Einstein gravity, the Bekenstein-Hawking relation S = A/(4G) defines the horizon entropy, where A = 4[pi][[??].sup.2.sub.A] is the area of the

apparent horizon [55].

Then one can show the following relation for the

apparent horizon:

Here, we are interested in investigating the effects of the displacement vector field on the

apparent horizon entropy of the FRW universe and thus its thermodynamics.