  Thermal transport in condensed matter systems is traditionally formulated as
a response to a background gravitational field. In this work, we seek a
twisted-boundary-condition formalism for thermal transport in analogy to the
$U(1)$ twisted boundary condition for electrical transport. Specifically, using
the transfer matrix formalism, we introduce what we call the energy-twisted
boundary condition, and study the response of the system to the boundary
condition. As specific examples, we obtain the thermal Meissner stiffness of
(1+1)-dimensional CFT, the Ising model, and disordered fermion models. We also
identify the boost deformation of integrable systems as a bulk counterpart of
the energy-twisted boundary condition. We show that the boost deformation of
the free fermion chain can be solved explicitly by solving the inviscid Burgers
equation. We also discuss the boost deformation of the XXZ model, and its
nonlinear thermal Drude weights, by studying the boost-deformed Bethe ansatz
equations.
