The minus sign in the first law of de Sitter horizons

Understand

Due to a well-known, but curious, minus sign in the Gibbons-Hawking first law for the static patch of de Sitter space, the entropy of the cosmological horizon is reduced by the addition of Killing energy.

• This minus sign raises the puzzling question how the thermodynamics of the static patch should be understood.
• We argue the confusion arises because of a mistaken interpretation of the matter Killing energy as the total internal energy, and resolve the puzzle by introducing a system boundary at which a proper thermodynamic ensemble can be specified.
• When this boundary shrinks to zero size the total internal energy of the ensemble (the Brown-York energy) vanishes, as does its variation.