In many engineering problems, the propagation of cracks is not only driven by mechanical load but also other physical phenomena, for instance, thermal effect, chemical reaction, and electromagnetic force. The solving strategies for above multiphysics problems can be classified into three categories, the monolithic scheme, the fully staggered scheme, and the partially staggered (hybrid) scheme. In the context of thermomechanically driven crack propagation, we present a highly efficient limited-memory BFGS (L-BFGS) method as a monolithic scheme for phase-field crack modeling. This monolithic scheme is well suited for solving fully coupled problems involving multiple fields, in this case, the displacement field, the temperature field, and the phase-field (damage field). An adaptive mesh refinement strategy is further developed to properly enforce the damage irreversibility based on the element-wise L2-projection of the history variable field. Two alternative solving schemes, including the fully staggered scheme and the partially staggered (hybrid) scheme, are provided for comparison. The fully staggered scheme partitions the coupled thermomechanical crack problem into three sub-problems: the mechanical sub-problem, the thermal sub-problem, and the phase-field sub-problem. The partially staggered (hybrid) scheme partitions the same crack problem into two sub-problems, the thermomechanical sub-problem and the phase-field sub-problem. For the first time, the performances of the developed L-BFGS monolithic scheme, the fully staggered scheme, and the partially staggered scheme are thoroughly compared under the same computational setup through a series of widely adopted benchmark problems. The impacts of various direct and iterative linear solvers are also discussed. The developed L-BFGS monolithic scheme offers a universal strategy to efficiently and robustly solve other types of multiphysics phase-field crack problems.