This paper investigates the thermodynamic properties of black hole horizons and cosmological horizons in Gauss-Bonnet gravity within de Sitter (dS) space. The authors study the solutions and thermodynamics of these horizons for both positive and negative Gauss-Bonnet coefficients. Key findings include: 1. **Black Hole Horizons:** - For a positive Gauss-Bonnet coefficient, a locally stable small black hole appears in five dimensions but disappears in dimensions $d \geq 6$. The black hole is always unstable quantum mechanically in these dimensions. - The cosmological horizon is always locally stable, but the pure de Sitter space is globally preferred due to its lower free energy. 2. **Cosmological Horizon:** - The cosmological horizon is always locally stable with positive heat capacity. - The pure de Sitter space is globally preferred, as evidenced by the positive difference in free energies between the cosmological horizon and the vacuum dS space. 3. **Negative Gauss-Bonnet Coefficient:** - There is a constraint on the Gauss-Bonnet coefficient, beyond which the gravity theory is not well-defined. - The horizon radius of the Gauss-Bonnet black hole has an upper bound and a lower bound, with the lower bound being more stringent. - The black hole horizon is always thermodynamically unstable, while the cosmological horizon remains stable. - The pure de Sitter space is still globally preferred. 4. **Conclusion:** - Both thermodynamic discussions lead to the conclusion that a pure de Sitter space is globally preferred, consistent with the dS/CFT correspondence where a pure de Sitter space corresponds to an UV fixed point of the dual field theory. The study provides insights into the thermodynamic properties of black hole horizons and cosmological horizons in Gauss-Bonnet gravity within dS space, highlighting the stability and phase structure of these horizons.This paper investigates the thermodynamic properties of black hole horizons and cosmological horizons in Gauss-Bonnet gravity within de Sitter (dS) space. The authors study the solutions and thermodynamics of these horizons for both positive and negative Gauss-Bonnet coefficients. Key findings include: 1. **Black Hole Horizons:** - For a positive Gauss-Bonnet coefficient, a locally stable small black hole appears in five dimensions but disappears in dimensions $d \geq 6$. The black hole is always unstable quantum mechanically in these dimensions. - The cosmological horizon is always locally stable, but the pure de Sitter space is globally preferred due to its lower free energy. 2. **Cosmological Horizon:** - The cosmological horizon is always locally stable with positive heat capacity. - The pure de Sitter space is globally preferred, as evidenced by the positive difference in free energies between the cosmological horizon and the vacuum dS space. 3. **Negative Gauss-Bonnet Coefficient:** - There is a constraint on the Gauss-Bonnet coefficient, beyond which the gravity theory is not well-defined. - The horizon radius of the Gauss-Bonnet black hole has an upper bound and a lower bound, with the lower bound being more stringent. - The black hole horizon is always thermodynamically unstable, while the cosmological horizon remains stable. - The pure de Sitter space is still globally preferred. 4. **Conclusion:** - Both thermodynamic discussions lead to the conclusion that a pure de Sitter space is globally preferred, consistent with the dS/CFT correspondence where a pure de Sitter space corresponds to an UV fixed point of the dual field theory. The study provides insights into the thermodynamic properties of black hole horizons and cosmological horizons in Gauss-Bonnet gravity within dS space, highlighting the stability and phase structure of these horizons.