Extreme Ultraviolet Explorer Deep Survey observations of cool stars (spectral type F to M) have been used to investigate the distribution of coronal flare rates in energy and its relation to activity indicators and rotation parameters. Cumulative and differential flare rate distributions were constructed and fitted with different methods. Power laws are found to approximately describe the distributions. A trend toward flatter distributions for later type stars is suggested in our sample. Assuming that the power laws continue below the detection limit, we have estimated that the superposition of flares with radiated energies of about 10²⁹-10³¹ ergs could explain the observed radiative power loss of these coronae, while the detected flares are contributing only ≈10%. Although the power-law index is not correlated with rotation parameters (rotation period, projected rotational velocity, Rossby number) and only marginally with the X-ray luminosity, the flare occurrence rate is correlated with all of them. The occurrence rate of flares with energies larger than 10³² ergs is found to be proportional to the average total stellar X-ray luminosity. Thus, energetic flares occur more often in X-ray bright stars than in X-ray faint stars. The normalized occurrence rate of flares with energies larger than 10³² ergs increases with increasing L_X/L_bol and stays constant for saturated stars. A similar saturation is found below a critical Rossby number. The findings are discussed in terms of simple statistical flare models in an attempt to explain the previously observed trend for higher average coronal temperatures in more active stars. It is concluded that flares can contribute a significant amount of energy to coronal heating in active stars.