From (29), it can be seen that the first law of thermodynamics is showing the validity when [GAMMA] = 3H(1 - (1/[gamma])[(3[H.sup.2] - [c.sup.2]/2[a.sup.6]).sup.-1] (([gamma]([eta] + 8[H.sup.2])/8[H.sup.2]) (3[H.sup.2] - [c.sup.2]/2[a.sup.6]) - [c.sup.2]/[a.sup.6])).

(iii) Power law corrected entropy: for power law corrected entropy, we have investigated that first law of thermodynamics holds at apparent horizon for [GAMMA] = 3H(1 - (1/[gamma]) [(3[H.sup.2] - [c.sup.2]/2[a.sup.6]).sup.-1]([gamma](3[H.sup.2] - [c.sup.2]/2[a.sup.6])) [(1/[L.sup.2.sub.p] - (2 - [delta]/2)([K.sub.[delta]]/[L.sup.2.sub.p]) [(1/H).sup.2-[delta]]).sup.-1] - [c.sup.2]/[a.sup.6]).

To investigate the universality of the redefinition of Hawking temperature on the event horizon, one can investigate the validity of the first law of thermodynamics. During an infinitesimal time interval, one can write the energy flux across the event horizon as [24-26, 28, 29]

From the First Law of Thermodynamics into Friedmann Equations in Palatini f(R) Gravity

Due to the presence of this extra term, the field equations do not obey the universal form of first law of thermodynamics dE = TdS + WdV in this gravity.

We have found that the total entropy in the first law of thermodynamics involves contribution from horizon entropy in terms of area and the entropy production term.

In this paper, we restudied Padmanabhan's work that it is possible to write Einstein's equation for spherically symmetric space-time in the form of the

first law of thermodynamics [14-17], but the thermodynamic quantities might not be consistent with the normal ones, especially the pressure and internal energy.

In this section, we will first calculate the thermodynamic quantities of the black hole obtained in the last section, then check the first law of thermodynamics, and finally extend the phase space to explore the phase transition behaviors of the black hole.

Now with all the thermodynamic quantities above in hand, one can check that the first law of thermodynamics

First Law of Thermodynamics. Now we investigate the thermodynamic behavior of the nonminimal f(T) gravity on the apparent horizon.

Consequently, the first law of thermodynamics can be expressed as follows:

Now, using these results as well as the

first law of thermodynamics (Td[S.sub.A] = pdV + dE), we obtain