  A stopping time $T$ is the first time when a trajectory of a stochastic
process satisfies a specific criterion. In this paper, we use martingale theory
to derive the integral fluctuation relation $\langle e^{-S_{\rm
tot}(T)}\rangle=1$ for the stochastic entropy production $S_{\rm tot}$ in a
stationary physical system at stochastic stopping times $T$. This fluctuation
relation implies the law $\langle S_{\rm tot}(T)\rangle\geq 0$, which states
that it is not possible to reduce entropy on average, even by stopping a
stochastic process at a stopping time, and which we call the second law of
thermodynamics at stopping times. This law implies bounds on the average amount
of heat and work a system can extract from its environment when stopped at a
random time. Furthermore, the integral fluctuation relation implies that
certain fluctuations of entropy production are universal or are bounded by
universal functions. These universal properties descend from the integral
fluctuation relation by selecting appropriate stopping times: for example, when
$T$ is a first-passage time for entropy production, then we obtain a bound on
the statistics of negative records of entropy production. We illustrate these
results on simple models of nonequilibrium systems described by Langevin
equations and reveal two interesting phenomena. First, we demonstrate that
isothermal mesoscopic systems can extract on average heat from their
environment when stopped at a cleverly chosen moment and the second law at
stopping times provides a bound on the average extracted heat. Second, we
demonstrate that the average efficiency at stopping times of an autonomous
stochastic heat engines, such as Feymann's ratchet, can be larger than the
Carnot efficiency and the second law of thermodynamics at stopping times
provides a bound on the average efficiency at stopping times.
