Thermodynamical study on universal horizons in higher $D$-dimensional spacetime and aether waves

We investigate thermodynamic behaviors of the $D$-dimensional gravity coupled to a dynamical unit timelike vector, the aether, present two kinds of exact charged solutions, and study the linearized wave spectrum of this theory. There is a universal horizon behind the Killing horizon in these aether black holes. It is interesting that there exist $D$-dimensional Schwarzschild, Reissner-Nordström, and charged Schwarzschild black holes but now with an universal horizon inside. We find that in the uncharged case for both kinds, or charged case but the charge $\overline{Q}\ll {\overline{r}}_0/2$ for the first kind, one can construct a Smarr formula and the (slightly modified) first law of black hole mechanics at the universal horizons for the aether black holes. An entropy can be associated with the universal horizon and a temperature can be defined there. For the second kind of aether black hole and for the first kind in the extremal higher dimensions, the work term involving charge $V_{UH}\delta Q$ disappeared. For the aether wave, our results show that the spin-1 and spin-2 modes are the same as those in 4-dimensional spacetime, and only the spin-0 one is different and dependant on the dimension number $n$.