Four-charge static non-extremal black holes in the five-dimensional $\mathcal{N}=2$, $STU-W^{2}U$ supergravity

We construct, for the first time, new static non-extremal five-dimensional black hole solutions (without or with squashed horizons) endowing with four different electric charge parameters in the $D=5$, $\mathcal{N}=2$ supergravity coupled to three vector multiplets with a specific pre-potential $\mathcal{V}=STU-W^{2}U\equiv 1$. When the fourth charge parameter disappears, the solution simplify reduces to the three-charge static black hole solution previously presented in ref. [1], which belongs to the solution to the $D=5$, $\mathcal{N}=2$ supergravity coupled to two vector multiplets (also notably known as the $STU$ model). We parameterize the model in such a simple fashion that not only can one easily recover the static three-charge solution but also it is very convenient to study their thermodynamical properties of the obtained black hole solutions in the case without a squashing horizon. We then show that the thermodynamical quantities perfectly obey both the differential first law and integral Smarr formula of thermodynamics. Finally, we also extend to present its generalizations with squashed horizons or including a nonzero cosmological constant.