This paper studies the thermodynamic properties of de Sitter (dS) space with a timelike boundary, focusing on static and spherically symmetric configurations in three and four spacetime dimensions. The authors impose conformal boundary conditions, fixing the conformal class of the induced metric and the trace of the extrinsic curvature at the boundary. They analyze the thermodynamic stability of dS space under these conditions, finding that for sufficiently large values of the extrinsic curvature trace K, the dS static patch is thermally stable. In contrast, the Dirichlet problem, where the induced metric is fixed, leads to negative specific heat for the region encompassing the cosmological horizon. In three dimensions, the analysis shows that the cosmic patch has positive specific heat for all values of K, and there is a phase transition at a critical inverse temperature. The conformal energy and entropy are derived, and a c-function is identified that decreases monotonically from the worldline limit to the stretched horizon limit. In four dimensions, the analysis reveals that the cosmic and black hole patches have different specific heats, with the cosmic patch having positive specific heat for certain values of K. The paper also explores the linearized dynamics of the dS space under conformal boundary conditions, finding that the modes correspond to quasinormal modes of the static patch and fluid dynamical modes near the cosmological horizon. The results highlight the importance of conformal boundary conditions in understanding the thermodynamics and stability of dS space.This paper studies the thermodynamic properties of de Sitter (dS) space with a timelike boundary, focusing on static and spherically symmetric configurations in three and four spacetime dimensions. The authors impose conformal boundary conditions, fixing the conformal class of the induced metric and the trace of the extrinsic curvature at the boundary. They analyze the thermodynamic stability of dS space under these conditions, finding that for sufficiently large values of the extrinsic curvature trace K, the dS static patch is thermally stable. In contrast, the Dirichlet problem, where the induced metric is fixed, leads to negative specific heat for the region encompassing the cosmological horizon. In three dimensions, the analysis shows that the cosmic patch has positive specific heat for all values of K, and there is a phase transition at a critical inverse temperature. The conformal energy and entropy are derived, and a c-function is identified that decreases monotonically from the worldline limit to the stretched horizon limit. In four dimensions, the analysis reveals that the cosmic and black hole patches have different specific heats, with the cosmic patch having positive specific heat for certain values of K. The paper also explores the linearized dynamics of the dS space under conformal boundary conditions, finding that the modes correspond to quasinormal modes of the static patch and fluid dynamical modes near the cosmological horizon. The results highlight the importance of conformal boundary conditions in understanding the thermodynamics and stability of dS space.