Topological Hochschild homology gives rise to a number of intrinsic derived invariants of a variety in positive characteristic. We will explain how to compute these quantities in terms of more classical invariants of X, such as de Rham and crystalline cohomology. As a consequence, we obtain new restrictions on the Hodge numbers of derived equivalent varieties in positive characteristic. We will also present an example of two derived equivalent 3-folds in characteristic 3 with different Hodge numbers. This is joint work with Benjamin Antieau and Nick Addington.