Phase Transitions and Critical Phenomena for the FRW Universe in an Effective Scalar-Tensor Theory

We find phase transitions and critical phenomena of the FRW (Friedmann-Robertson-Walker) universe in the framework of an effective scalar-tensor theory that belongs to the Horndeski class. We identify the thermodynamic pressure (generalized force) $P$ of the FRW universe in this theory with the work density $W$ of the perfect fluid, which is a natural definition directly read out from the first law of thermodynamics. We derive the thermodynamic equation of state $P=P(V, T)$ for the FRW universe in this theory and make a thorough discussion of its $P$-$V$ phase transitions and critical phenomena. We calculate the critical exponents, and show that they are the same with the mean field theory, and thus obey the scaling laws.