Does spatial curvature generate new thermodynamic criticality at the FLRW apparent horizon?

We generalize the apparent-horizon thermodynamic construction of Ref. [1] to a FLRW universe with non-zero spatial curvature, non-interacting cold dark matter and holographic-type dark energy. For nonzero curvature, the scale factor is an additional geometric variable in the horizon equation of state. The usual criticality conditions are thus not well defined until a closure prescription for the curvature sector is provided. We introduce a dimensionless curvature variable and restrict the thermodynamic variations to slices of constant value of this variable. For each such slice, a positive holographic coupling and nonlinear powers larger than one guarantee the existence of a unique positive critical point. The linear case n = 1 is excluded from the critical branch: the competing inverse powers in the equation of state then coalesce, and the critical formulas ( n − 1 ) − 1 enter a singular limiting regime rather than defining an isolated positive-temperature critical point. The critical specific volume, temperature and pressure are shifted by the spatial curvature, while the critical ratio and the mean-field critical exponents remain unchanged. It thus rescales the critical quantities, without generating a new local universality class. We construct an effective Helmholtz potential out of the physical thermodynamic volume and the entropy conjugate to the rescaled horizon temperature. In the quadratic holographic model, the parametric coexistence curve which is equivalent to the Maxwell construction is derived from the equality of the Gibbs free energies of the competing branches. The associated latent heat vanishes at the critical endpoint. These results establish first-order equilibrium coexistence within the effective fixed-curvature thermodynamic model, but do not by themselves demonstrate a phase transition realized along a cosmological trajectory.