  We consider thermodynamically consistent autonomous Markov jump processes
displaying a macroscopic limit in which the logarithm of the probability
distribution is proportional to a scale-independent rate function (i.e., a
large deviations principle is satisfied). In order to provide an explicit
expression for the probability distribution valid away from equilibrium, we
propose a linear response theory performed at the level of the rate function.
We show that the first order non-equilibrium contribution to the steady state
rate function, $g(x)$, satisfies $u(x)\cdot \nabla g(x) = -\beta \dot W(x)$
where the vector field $u(x)$ defines the macroscopic deterministic dynamics,
and the scalar field $\dot W(x)$ equals the rate at which work is performed on
the system in a given state $x$. This equation provides a practical way to
determine $g(x)$, significantly outperforms standard linear response theory
applied at the level of the probability distribution, and approximates the rate
function surprisingly well in some far-from-equilibrium conditions. The method
applies to a wealth of physical and chemical systems, that we exemplify by two
analytically tractable models - an electrical circuit and an autocatalytic
chemical reaction network - both undergoing a non-equilibrium transition from a
monostable phase to a bistable phase. Our approach can be easily generalized to
transient probabilities and non-autonomous dynamics. Moreover, its recursive
application generates a virtual flow in the probability space which allows to
determine the steady state rate function arbitrarily far from equilibrium.
