This article introduces a weight function method for fracture analysis of a circumferentially cracked functionally graded hollow cylinder subjected to transient thermomechanical loading. Analytical solutions for transient temperature and stress distributions in the uncracked cylinder are derived by applying finite Hankel transformation. These solutions are utilized to determine stress acting on the faces of the circumferential crack in the local perturbation problem. Thermomechanical material properties are assumed to be power functions of the radial coordinate in the derivations. Coefficients of the weight function are found using reference stress intensity factors computed through the finite element method. Domain form of the J-integral is used in the finite element calculations. Comparisons of the numerical results calculated by the proposed weight function method to those generated by finite element analysis demonstrate the high level of accuracy attained by the application of the developed procedures. Further parametric analyses are presented to illustrate the influences of dimensionless time, crack depth to thickness ratio, power law index, and convection coefficient upon transient mode I thermomechanical stress intensity factors.