= Solution
Use units $G=\hbar=c=k_B=1$ and put $\ell=\sqrt{3/\Lambda}$. The <static patch of de Sitter spacetime> has radial factor $f=1-r^2/\ell^2$. A <Wick rotation> $t=-i\tau$ makes its radial-time metric positive:
$$
ds_E^2=f\,d\tau^2+\frac{dr^2}{f}+r^2d\Omega_2^2.
$$
Near the <cosmological horizon>, write $r=\ell-\rho^2/(2\ell)$. Then $f=\rho^2/\ell^2+O(\rho^4)$ and
$$
ds_E^2=d\rho^2+\rho^2d(\tau/\ell)^2+\ell^2d\Omega_2^2+O(\rho^2)
$$
in the radial-time directions. Thus $\tau/\ell$ must be an angle of period $2\pi$. A larger integer multiple would give a conical excess rather than a smooth origin. The <Euclidean black-hole regularity condition> gives
$$
\boxed{\beta=2\pi\ell=2\pi\sqrt{3/\Lambda},\qquad T=\beta^{-1}=\frac1{2\pi}\sqrt{\Lambda/3}.}
$$
This <de Sitter horizon temperature> uses the time normalization at $r=0$, where $t$ is <proper time>; static observers at other radii see the corresponding <Tolman temperature law> redshift.
Vary the gravitational <Euclidean action> with respect to $g^{ab}$. The identities $\delta\sqrt g=-\tfrac12\sqrt g\,g_{ab}\delta g^{ab}$ and the integrated variation of the <scalar curvature> yield, after discarding the assumed boundary term,
$$
\delta I_E=-\frac1{16\pi}\int\sqrt g\,\left(R_{ab}-\frac12Rg_{ab}+\Lambda g_{ab}\right)\delta g^{ab}\,d^4x.
$$
The resulting vacuum <Einstein field equations> are therefore
$$
\boxed{R_{ab}-\frac12Rg_{ab}+\Lambda g_{ab}=0.}
$$
Tracing in four dimensions gives $R=4\Lambda$, and substitution gives $R_{ab}=\Lambda g_{ab}$.
The smooth Euclidean geometry is compact. Explicitly, in Euclidean five-space use
$$
X_0=\sqrt{\ell^2-r^2}\cos(\tau/\ell),\quad
X_1=\sqrt{\ell^2-r^2}\sin(\tau/\ell),\quad
(X_2,X_3,X_4)=r\,\mathbf n,\quad |\mathbf n|=1.
$$
These obey $\sum X_A^2=\ell^2$ and induce precisely the above metric, so $0\leq r\leq\ell$ with the identified Euclidean time covers a round four-sphere once. Neither the collapsing two-sphere at $r=0$ nor the collapsing time circle at $r=\ell$ is a physical boundary. Since $\sqrt g=r^2\sin\theta$, its volume and on-shell <Euclidean action> are
$$
\operatorname{Vol}=\beta\,4\pi\int_0^\ell r^2dr=\frac{8\pi^2\ell^4}{3}=\frac{24\pi^2}{\Lambda^2},
\qquad
\boxed{I_E=-\frac{2\Lambda}{16\pi}\operatorname{Vol}=-\frac{3\pi}{\Lambda}.}
$$
The semiclassical <partition function> is $Z\simeq e^{-I_E}$, and $I_E=\beta F=\beta E-S$ follows from the <free energy> relation $F=E-TS$. With the stipulated zero internal energy,
$$
\boxed{S=-I_E=\frac{3\pi}{\Lambda}=\frac{A_H}{4},\qquad A_H=4\pi\ell^2.}
$$
Thus the <Euclidean de Sitter action and entropy> reproduce the <Bekenstein-Hawking entropy> of the <cosmological horizon> without a boundary contribution. Restoring constants, $S=k_Bc^3A_H/(4G\hbar)$.
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