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Gravitational theories invariant under transverse diffeomorphisms and Weyltransformations have the same classical solutions as the corresponding fullydiffeomorphism invariant theories. However, they solve some of the problemsrelated to the cosmological constant and in principle allow local energynon-conservation. In the present work, we obtain the Noether charge formalismfor these theories. We first derive expressions for the Noether currents andcharges corresponding to transverse diffeomorphisms and Weyl transformations,showing that the latter vanish identically. We then use these results to obtainan expression for a perturbation of a Hamiltonian corresponding to evolutionalong a transverse diffeomorphism generator. From this expression, we derivethe first law of black hole mechanics, identifying the total energy, the totalangular momentum, the Wald entropy, and the contributions of the cosmologicalconstant perturbations and energy non-conservation. Lastly, we extend ourformalism to derive the first law of causal diamonds.

Noether charge formalism for Weyl invariant theories of gravity