Aims. Variability at all observed wavelengths is a distinctive property of active galactic nuclei (AGN). Hard X-rays provide us with a view of the innermost regions of AGN, mostly unbiased by absorption along the line of sight. Characterizing the intrinsic hard X-ray variability of a large AGN sample and comparing it to the results obtained at lower X-ray energies can significantly contribute to our understanding of the mechanisms underlying the high-energy radiation. Methods. Swift/BAT provides us with the unique opportunity to follow, on time scales of days to years and with regular sampling, the 14–195 keV emission of the largest AGN sample available up to date for this kind of investigation. As a continuation of an early work using the first 9 months of BAT data, we study the amplitude of the variations and their dependence on subclass and on energy, for a sample of 110 radio quiet and radio loud AGN selected from the BAT 58-month survey. Results. About 80% of the AGN in the sample are found to exhibit significant variability on month-to-year time scales. In particular, radio loud sources are the most variable, and Seyfert 1.5−2 galaxies are slightly more variable than Seyfert 1, while absorbed and unabsorbed objects show similar timing properties. The amplitude of the variations and their energy dependence are incompatible with variability being driven at hard X-rays by changes in the absorption column density. In general, the variations in the 14–24 and 35–100 keV bands are correlated well, suggesting a common origin to the variability across the BAT energy band. However, radio quiet AGN display on average 10% larger variations at 14–24 keV than at 35–100 keV, and a softer-when-brighter behavior for most of the Seyfert galaxies with detectable spectral variability on a time scale of a month. In addition, sources with harder spectra are found to be more variable than softer ones, unlike what it is observed below 10 keV. These properties are generally consistent with a variable, in flux and shape, power law continuum, pivoting at energies ≳ 50 keV, to which a constant reflection component is superposed. When the same time scales are considered, the timing properties of AGN at hard X-rays are comparable to those at lower energies, with at least some of the differences possibly ascribable to components contributing differently in the two energy domains (e.g., reflection, absorption).