In this paper, we use Born-Infeld black holes to test two recent holographic conjectures of complexity, the "Complexity = Action" (CA) duality and "Complexity = Volume 2.0" (CV) duality.

We investigate the entanglement temperature of a small scale subsystem in low excited states by using holographic method. Especially, we study the entanglement entropy and entanglement temperature in higher derivative gravities which are considered as low thermal excitation of pure AdS gravity. We find that the entanglement entropy are related to the central charges of CFT living on the boundary. The relation between the variance of entanglement entropy and energy of a small scale subsystem has been also obtained. Furthermore, the relation is consistent with the first law-like relation that is proposed by Phys. Rev. Lett. 110, 091602 (2013). Finally, we derive the formula of the variance of entanglement entropy in general excited states in gravity background with the Fefferman-Graham coordinates and the entanglement temperature can be figured out in special case.

In this paper, we study the thermodynamics and the weak cosmic censorship conjecture of the RN-AdS black hole surrounded by the quintessence under the scattering of a charged complex scalar field. With scalar field scattering, the variation of the black hole is calculated in the extended and normal phase spaces. In the extended phase space, the cosmological constant and the normalization parameter are considered as thermodynamic variables, and the first law of thermodynamics is valid, but the second law of thermodynamics is not valid. In the normal phase space, the cosmological constant and the normalization parameter are fixed, and the first and second laws of thermodynamics can also be satisfied. Furthermore, the weak cosmic censorship conjecture is both valid in the extended and normal phase spaces.