  We extend the finite-temperature Keldysh non-equilibrium coupled cluster
theory (Keldysh-CC) [{\it J. Chem. Theory Comput.} \textbf{2019}, 15,
6137-6253] to include a time-dependent orbital basis. When chosen to minimize
the action, such a basis restores local and global conservation laws
(Ehrenfest's theorem) for all one-particle properties, while remaining energy
conserving for time-independent Hamiltonians. We present the time-dependent
orbital-optimized coupled cluster doubles method (Keldysh-OCCD) in analogy with
the formalism for zero-temperature dynamics, extended to finite temperatures
through the time-dependent action on the Keldysh contour. To demonstrate the
conservation property and understand the numerical performance of the method,
we apply it to several problems of non-equilibrium finite-temperature dynamics:
a 1D Hubbard model with a time-dependent Peierls phase, laser driving of
molecular H$_2$, driven dynamics in warm-dense silicon, and transport in the
single impurity Anderson model.
