We examine the pressure and temperature dependences of a nanoscale structural rearrangement in periodic silica/surfactant composites with a goal of understanding how the silica framework modifies the molecular environment of the organic surfactant. Hydrothermal treatment is used to force surfactant-templated silica to undergo a phase transition from a hexagonal phase to a lamellar phase; the experiment is carried out both at 0.001 and at 1 GPa. The transition midpoint shifts to a higher temperature when the composite transforms under elevated pressure. Applying the concepts of surfactant packing and compressibility to this curvature-driven transition leads us to estimate the pressure felt by the composite's organic phase when the composite is subjected to an external pressure of 1 GPa. Our results indicate that this internal pressure is approximately 0.1 GPa, which is significantly smaller than the applied pressure of 1 GPa. The results confirm the idea that the high rigidity of the silica framework provides a protected molecular scale environment within the organic domains of the composite.