  We investigate thermodynamic aspects of black holes in the recently
formulated four dimensional Gauss-Bonnet theory of gravity, focusing on its
asymptotically de Sitter ($\Lambda>0$) solutions. We take a Euclidean path
integral approach, where thermodynamic data is fixed at a finite radius
`cavity' outside the black hole to achieve equilibrium in the presence of the
cosmological horizon. Working in the extended phase space where the
cosmological constant is treated as a thermodynamic pressure, we study the
phase structure of both uncharged and charged solutions, uncovering a wealth of
phenomena. In the uncharged case, black holes are found to undergo either the
standard Hawking-Page-like transition to empty de Sitter space or a small-large
transition (akin to those seen in charged AdS black holes in pure Einstein
gravity) depending on the pressure. We also find a triple point where the
radiation, small, and large black hole phases coexist, and a zeroth-order phase
transition within a narrow range of Gauss-Bonnet coupling parameter. Reentrant
phase transitions between radiation and a small black hole also exist inside a
certain parameter range. Similar phenomena are found in the charged case, with
the charge parameter playing a role analogous to the Gauss-Bonnet coupling
parameter in determining the phase structure.
