Recent discovery of a distinct type of topological insulating state in tin telluride (SnTe) has reignited great interest in this narrow bandgap semiconductor. A pressing task is to understand the stability of this intriguing state of matter and the response of its electronic properties under the influence of external conditions that may alter the underlying fundamental physics. In this work, we examine by first-principles calculations the effect of pressure on the phonon dispersion, electronic band structure, Fermi surface, and charge distribution of SnTe. We show that pressure suppresses a soft optical phonon mode and enhances the stability of the cubic (B1, Fm3̅m) phase of SnTe, which is a key requirement for the observed topological insulating state. Pressure also drives an electronic topological transition that alters the shape and connectivity of the Fermi surface, and it also promotes a charge redistribution that results in an enhanced bonding interaction that strengthens the cubic crystalline structure. Our calculations further demonstrate that the electronic band gap of the cubic phase, another key parameter of the topological insulating state in SnTe, increases in size monotonically with increasing pressure. These results indicate that pressure makes the topological insulating state in the cubic phase of SnTe more stable and robust; pressure is also effective in tuning the electronic band structure, which is expected to have significant impact on a wide range of physical properties crucial to potential applications.