We study the volume of compact Riemannian manifolds which are Einstein with respect to a metric connection with (parallel) skew-torsion. We provide a result for the sign of the first variation of the volume in terms of the corresponding scalar curvature. This generalizes a result of M. Ville [15] related with the first variation of the volume on a compact Einstein manifold.

• 53B05 - Linear and affine connections
• 53C05 - Connections (general theory)
• 53C25 - Special Riemannian manifolds (Einstein, Sasakian, etc.)