Let S be an oriented compact surface with negative Euler characteristic. Teichmüller space of S is the space of isotopy classes of marked hyperbolic structures on S. There is a well-known Riemannian metric on the Teichmüller space, called the Weil-Petersson metric. It has many interesting properties, for example: Kähler, negatively curved, incomplete, geodesically convex, etc. When S is closed, this metric can be interpreted as the pressure metric by using the thermodynamic formalism. When S has boundary, both of these two metrics are still well-defined but the relation between them are not known yet. By studying the incompleteness of the pressure metric in the second case, we are capable to answer that they are not equivalent to each other.