  Fluctuation theorems specify the non-zero probability to observe negative
entropy production, contrary to a naive expectation from the second law of
thermodynamics. For closed particle trajectories in a fluid, Stokes theorem can
be used to give a geometric characterization of the entropy production.
Building on this picture, we formulate a topological fluctuation theorem that
depends only by the winding number around each vortex core and is insensitive
to other aspects of the force. The probability is robust to local deformations
of the particle trajectory, reminiscent of topologically protected modes in
various classical and quantum systems. We demonstrate that entropy production
is quantized in these strongly fluctuating systems, and it is controlled by a
topological invariant. We demonstrate that the theorem holds even when the
probability distributions are non-Gaussian functions of the generated heat.
