  We study conserved charges and thermodynamics of analytic rotating anti-de
Sitter black holes with extended horizon topology -- also known as black
strings -- in dynamical Chern-Simons modified gravity. The solution is
supported by a scalar field with an axionic profile that depends linearly on
the coordinate that spans the string. We compute conserved charges by making
use of the renormalized boundary stress-energy tensor. Then, by adopting the
Noether-Wald formalism, we compute the black string entropy and obtain its area
law. Indeed, the reduced Euclidean Hamiltonian approach shows that these
methods yield a consistent first law of thermodynamics. Additionally, we derive
a Smarr formula using a radial conservation law associated to the scale
invariance of the reduced action and obtain a Cardy formula for the black
string. A first-order phase transition takes place at a critical temperature
between the ground state and the black string, above which the black string is
the thermodynamically favored configuration.
