  We demonstrate that the counting statistics of currents in periodically
driven ergodic stochastic systems can show sharp changes of some of its
properties in response to continuous changes of the driving protocol. To
describe this effect, we introduce a new topological phase factor in evolution
of the moment generating function which is akin to the topological geometric
phase in evolution of a periodically driven quantum mechanical system with
time-reversal symmetry. This phase leads to the prediction of a sign change for
the difference of the probabilities to find even and odd number particles
transferred in a stochastic system in response to cyclic evolution of control
parameters. The driving protocols that lead to such a sign change should
enclose specific degeneracy points in the space of control parameters. The
relation between topology of the paths in the control parameter space and the
sign changes can be described in terms of the first Stiefel-Whitney class.
