  We propose a vertex representation of the tensor network (TN) in the anti-de
Sitter space (AdS$_{2+0}$) that we model on a subset of hyperbolic lattices.
The tensors form a network of regular $p$-sided polygons ($p>4$) with the
coordination number four. The response to multi-state spin systems on the
hyperbolic TN is analyzed for their entire parameter space. We show that
entanglement entropy is sensitive to distinguish various hyperbolic geometries
whereas other thermodynamic quantities are not. We test the numerical accuracy
of vertex TNs in the phase transitions of the first, second, and infinite order
at the point of maximal entanglement entropy. The hyperbolic structure of TNs
induces non-critical properties in the bulk although boundary conditions
significantly affect the total free energy in the thermodynamic limit. Thus
developed vertex-type TN can be applied to study low-energy quantum states on
AdS.
