The paper explores the conformal boundary conditions in de Sitter space, focusing on the static patch subject to these conditions. The authors study the thermodynamic structure of de Sitter space in both three and four spacetime dimensions, imposing that the conformal class of the induced metric and the trace of the extrinsic curvature are fixed at the boundary. In three spacetime dimensions, they find that the spacetime is thermally stable for all values of the trace of the extrinsic curvature. In four spacetime dimensions, the thermal stability depends on the value of the trace of the extrinsic curvature, with sufficiently large values leading to thermal stability. The paper also analyzes the linearized Einstein equations under conformal boundary conditions, showing that modes in the worldline limit are related to quasinormal modes, and modes near the cosmological horizon are linked to shear and sound modes of a fluid dynamical system. Additionally, modes with positive imaginary frequencies grow polynomially in a local inertial frame. The results contrast those obtained from Dirichlet boundary conditions, where the specific heat of the cosmic patch is always negative.The paper explores the conformal boundary conditions in de Sitter space, focusing on the static patch subject to these conditions. The authors study the thermodynamic structure of de Sitter space in both three and four spacetime dimensions, imposing that the conformal class of the induced metric and the trace of the extrinsic curvature are fixed at the boundary. In three spacetime dimensions, they find that the spacetime is thermally stable for all values of the trace of the extrinsic curvature. In four spacetime dimensions, the thermal stability depends on the value of the trace of the extrinsic curvature, with sufficiently large values leading to thermal stability. The paper also analyzes the linearized Einstein equations under conformal boundary conditions, showing that modes in the worldline limit are related to quasinormal modes, and modes near the cosmological horizon are linked to shear and sound modes of a fluid dynamical system. Additionally, modes with positive imaginary frequencies grow polynomially in a local inertial frame. The results contrast those obtained from Dirichlet boundary conditions, where the specific heat of the cosmic patch is always negative.