Holographic screens are the generalization of the event horizon of a black hole in entropic force scheme, which are defined by setting Newton potential $phi$ constant, textit{i. e.} $e^{2phi}=c=$const. By demonstrating that the integrated first law o
f thermodynamics is equivalent to the ($r-r$) component of Einstein equations, We strengthen the correspondence between thermodynamics and gravity. We show that there are not only the first law of thermodynamics, but also kinds of phase transitions of holographic screens. These phase transitions are characterized by the discontinuity of their heat capacities. In (n+1) dimensional Reissner-Nordstr{o}m-anti-de Sitter (RN-AdS) spacetime, we analyze three kinds of phase transitions, which are of the holographic screens with Q=0 (charge), constant $Phi$ (electrostatic potential) and non-zero constant $Q$. In the Q=0 case, only the holographic screens with $0le c<1$ can undergo phase transition. In the constant $Phi$ case, the constraints become as $0le c+16tilde{Gamma}^{2}Phi^{2}<1$, where $tilde{Gamma}$ is a dimensional dependent parameter. By verifying the Ehrenfest equations, we show that the phase transitions in this case are all second order phase transitions. In the constant $Q$ case, there might be two, or one, or no phase transitions of holographic screens, depending on the values of $Q$ and $c$.
We study the property of matter in equilibrium with a static, spherically symmetric black hole in D-dimensional spacetime. It requires this kind of matter has an equation of state (omegaequiv p_r/rho=-1/(1+2kn), k,nin mathbb{N}), which seems to be in
dependent of D. However, when we associate this with specific models, some interesting limits on space could be found: (i)(D=2+2kn) while the black hole is surrounded by cosmic strings; (ii)the black hole can be surrounded by linear dilaton field only in 4-dimensional spacetime. In both cases, D=4 is special.
