In this paper, we investigate the thermodynamics especially the Hawking-Page-like phase transition of the McVittie space-time. We formulate the first law of thermodynamics for the McVittie black hole, and find that the work density W of the perfect fluid plays the role of the thermodynamic pressure, i.e. P:=-W. We also construct the thermodynamic equation of state for the McVittie black hole. Most importantly, by analysing the Gibbs free energy, we find that the Hawking-Page-like phase transition from FRW spacetime to McVittie black hole is possible in the case P > 0.

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The Hamiltonian analysis for a 3-dimensional connection dynamics of so(1, 2), spanned by {L−+, L−2, L+2} instead of {L01, L02, L12}, is first conducted in a Bondi-like coordinate system. The symmetry of the system is clearly presented. A null coframe with 3 independent variables and 9 connection coefficients are treated as basic configuration variables. All constraints and their consistency conditions, the solutions of Lagrange multipliers as well as the equations of motion are presented. There is no physical degree of freedom in the system. The Bañados-Teitelboim-Zanelli (BTZ) spacetime is discussed as an example to check the analysis. Unlike the ADM formalism, where only non-degenerate geometries on slices are dealt with and the Ashtekar formalism, where non-degenerate geometries on slices are mainly concerned though the degenerate geometries may be studied as well, in the present formalism the geometries on the slices are always degenerate though the geometries for the spacetime are not degenerate.

We define thermodynamic pressure P by work density W as the conjugate quantity of thermodynamic volume V from field equation. We derive the equations of state P=P(V, T) for the Friedmann–Robertson–Walker (FRW) universe in Einstein gravity and a modified gravity respectively. We find that the equation of state from Einstein gravity shows no P-V phase transition, while the equation of state from the modified gravity does, where the critical exponents are the same as those in mean field theory.

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The smooth 1-form Verma module of so(1, 4) is acquired.

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