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This paper presents the mathematical framework and the asynchronous finite element solver that captures the brittle fractures in multi-phase fluid-infiltrating porous media at the mesoscale where the constituents are not necessarily in a thermal equilibrium state. To achieve this goal, we introduce a dual-temperature effective medium theory in which the distinct constituent temperatures are homogenized independently whereas the heat exchange among the constituents is captured via phenomenological heat exchange laws in analog to the dual-permeability theory. To handle the different growth rates of the boundary layers in a stable and computationally efficient manner, an asynchronous time integrator is proposed and implemented in an operator-split algorithm that updates the displacement, pore pressure, phase field, and temperature of each constituent in an asynchronous manner. Numerical examples are introduced to verify the implementation and compare the path-dependent behaviors predicted by the dual-temperature and one-temperature models. [ABSTRACT FROM AUTHOR]