  The covariant Noether charge formalism (also known as the covariant phase
method) of Wald and collaborators, including its cohomological extension, is a
manifestly covariant Hamiltonian formalism that, in principle, allows one to
define and compute the energy, angular momenta, and chemical potentials of
generic solutions of gravitational theories. However, it has been observed that
for some supergravity solutions the variation of the Noether charge is not (at
least manifestably) integrable, and as a result it is unclear whether there are
well-defined thermodynamic charges for these solutions. In this work, we derive
an expression for the variation of the covariant Noether charges for any
solution of Type IIB 10-dimensional supergravity or 11-dimensional
supergravity. Although this Noether quantity is not integrable in general, we
show that for asymptotically scale-invariant solutions, it is. In particular,
the asymptotic scale-invariance allows one to define an energy density and
conjugate chemical potentials which obey a first law of thermodynamics and a
Smarr relation. These two thermodynamic relations can then be shown to imply
that the variation of the Noether charge is integrable, and accordingly the
energy and other thermodynamic charges may be defined unambiguously. We
explicitly demonstrate and illustrate our claim by computing the thermodynamic
charges of two non-trivial supergravity solutions that were recently
constructed: 1) the Polchinski-Strassler black brane that is dual to the
deconfined phase of $\mathcal{N}=1^*$ theory, and 2) the CGLP black brane that
asymptotes to the mass deformed Cveti\v{c}-Gibbons-L\"u-Pope (CGLP) solution.
