Annular seals are used in turbomachinery to reduce secondary flow between regions of high and low pressure. In a vibrating rotor system, the nonaxisymmetric pressure field developed in the small clearance between the rotor and the seal generate reactionary forces that can affect the stability of the entire rotor system. Traditionally, two analyses have been used to study the fluid flow in seals, bulk-flow analysis and computational fluid dynamics (CFD). Bulk-flow methods are computational inexpensive, but solve simplified equations that rely on empirically derived friction factor coefficients and are moderately accurate. CFD analyses generally provide more accurate results than bulk-flow codes, but solution time can vary between days and weeks. For gas damper seals, these analyses have been developed with the assumption that the flow can be treated as isothermal. However, some experimental studies have shown that the temperature change across the seal can be as much as 37%. Thus, a comprehensive analysis requires the solution of an energy equation. Recently, a new hybrid method that employs a CFD analysis for the zeroth-order flow and a bulk-flow analysis for the first-order, perturbed flow has been developed. This method has shown to compare well with full CFD analysis and experimental data while being computationally efficient. In this study, the previously developed hybrid method is extended to include the effects of nonisothermal flow. The hybrid method with energy equation is then compared with the isothermal hybrid method and experimental data for several test cases of hole-pattern seals and the importance of the use of energy equation is studied.