High-temperature compressible turbulence is complex and is of relevance in flow fields such as the atmospheric re-entry of hypersonic vehicles or the supersonic combustion of air-breathing engines. In these high-temperature environments, various complex processes occur, including the excitation of vibrational energy, molecular dissociation, and ionization. The vibrational non-equilibrium may significantly affect various non-linear turbulent processes, such as scalar mixing, energy cascading, intermittency, and material element deformation. The dynamics of velocity gradients is key to examining and understanding the behavior of these non-linear processes within a turbulent flow field. Therefore, in this work, we first aim to derive the exact evolution equation of the velocity gradient and the pressure Hessian tensor in a state of vibrational non-equilibrium. The derived equation set is comprised of various non-local and mathematically unclosed processes. For a deeper exploration and better comprehension of the physics inherent in these mechanisms, we then perform direct numerical simulation (DNS) of a decaying homogeneous isotropic flow field over a range of Mach number, Reynolds number and Damköhler number. Specifically, we identify how these non-dimensional parameters affect the evolution of the pressure Hessian tensor and the velocity gradient tensor in a compressible flow field. Our findings reveal that, among these non-dimensional parameters, the Damköhler number predominantly influences the statistics governed by the velocity gradient and the pressure Hessian tensors in the flow field , while the other parameters, such as the Mach number, Reynolds number, or the initial ratio of vibrational to local temperature, have a comparatively lesser impact. Additionally, we observe a prominent suppression in the strength of the pressure Hessian Tensor when the vibrational relaxation process becomes more rapid. This, in turn, mitigates the steepening of the velocity gradient field and the dilatational fluctuations. Further, we explore the possibility of leveraging the understanding gained from this study to develop closure models for the pressure Hessian tensor in vibrationally-excited flow fields.