Euclidean de Sitter action and entropy ($I_E=-3\pi /\Lambda ,S=3\pi /\Lambda$)

In units $G=\hslash =c=k_B=1$, a smooth Wick rotation of the four-dimensional static patch of de Sitter spacetime has Euclidean time period $2\pi \sqrt{3/\Lambda }$. Its compact geometry is a round four-sphere of radius $\sqrt{3/\Lambda }$. The Einstein field equations give $R=4\Lambda$, and direct integration gives volume $24{\pi }^2/{\Lambda }^2$, hence gravitational Euclidean action $I_E=-3\pi /\Lambda$. With zero assigned internal energy, $I_E=\beta E-S$ gives Bekenstein-Hawking entropy $S=3\pi /\Lambda =A/4$.