  We study the thermodynamics of Kaluza-Klein black holes with momentum along
the compact dimension, but vanishing angular momentum. These black holes are
stationary, but non-rotating. We derive the first law for these spacetimes and
find that the parameter conjugate to variations in the length of the compact
direction is an effective tension, which generally differs from the ADM
tension. For the boosted black string, this effective tension is always
positive, while the ADM tension is negative for large boost parameter. We also
derive two Smarr formulas, one that follows from time translation invariance,
and a second one that holds only in the case of exact translation symmetry in
the compact dimension. Finally, we show that the `tension first law' derived by
Traschen and Fox in the static case has the form of a thermodynamic Gibbs-Duhem
relation and give its extension in the stationary, non-rotating case.
