  We consider how variations in the moduli of the compactification manifold
contribute pdV type work terms to the first law for Kaluza-Klein black holes.
We give a new proof for the circle case, based on Hamiltonian methods, which
demonstrates that the result holds for arbitrary perturbations around a static
black hole background. We further apply these methods to derive the first law
for black holes in 2-torus compactifications, where there are three real
moduli. We find that the result can be simply stated in terms of constructs
familiar from the physics of elastic materials, the stress and strain tensors.
The strain tensor encodes the change in size and shape of the 2-torus as the
moduli are varied. The role of the stress tensor is played by a tension tensor,
which generalizes the spacetime tension that enters the first law in the circle
case.
