HH-twisted Lie algebroids

We define a new kind of algebroid which fulfills a Leibniz rule, a Jacobi identity twisted by a 3-form HH with values in the kernel of the anchor map, and the twist is closed under a naturally occurring exterior covariant derivative. We give examples and define three kinds of cohomologies, two via realizations as QQ-structures on graded manifolds. The paper classifies PQPQ-manifolds of maximal degree 3.