  We present a temperature extrapolation technique for self-consistent
many-body methods, which provides a causal starting point for converging to a
solution at a target temperature. The technique employs the Carath\'eodory
formalism for interpolating causal matrix-valued functions and is applicable to
various many-body methods, including dynamical mean field theory, its cluster
extensions, and self-consistent perturbative methods such as the
self-consistent GW approximation. We show results that demonstrate that this
technique can efficiently simulate heating and cooling hysteresis at a
first-order phase transition, as well as accelerate convergence.
