A time-varying Koopman operator (TVKO) is presented to study the nonlinear coupled dynamics between structural dynamics, heat transfer, and hypersonic aerothermodynamics, viz. AeroThermoElasticity (ATE). TVKO can be considered as an extension of the classical time-invariant Koopman operator (TIKO). It utilizes a subspace realization method known as the time-varying Eigensystem Realization Algorithm (TVERA) to approximate the underlying nonlinear model as a time varying linear model in a lifting space from time histories of input-output data. A benchmark model depicting the complex ATE dynamics for the flutter of a heated panel is considered to show the efficacy of the presented approach. The numerical experiments performed demonstrate the accuracy of the presented approach in capturing the bifurcation behavior in limit cycle oscillations due to variations in dynamic pressure.