Active matter describes systems whose constituents convert energy from their surroundings into directed motion, such as bacteria or catalytic colloids. We discover a thermodynamic law for suspensions of such active particles in the form of a remarkable direct link between their dynamics and thermodynamics: The irreversibility is a function of the thermodynamic system parameters, most importantly the swim pressure. Like in the famous fluctuation theorems of stochastic thermodynamics, irreversibility is a dynamical measure that quantifies the amount by which microscopic particle trajectories break time-reversal symmetry. Our discovery is based on numerical simulations and on analytical arguments for the limits of low particle density and/or high activity in the paradigmatic model of active Ornstein-Uhlenbeck particles.