  We construct the CFT dual of the first law of spherical causal diamonds in
three-dimensional AdS spacetime. A spherically symmetric causal diamond in
AdS$_3$ is the domain of dependence of a spatial circular disk with vanishing
extrinsic curvature. The bulk first law relates the variations of the area of
the boundary of the disk, the spatial volume of the disk, the cosmological
constant and the matter Hamiltonian. In this paper we specialize to first-order
metric variations from pure AdS to the conical defect spacetime, and the bulk
first law is derived following a coordinate based approach. The AdS/CFT
dictionary connects the area of the boundary of the disk to the differential
entropy in CFT$_2$, and assuming the `complexity=volume' conjecture, the volume
of the disk is considered to be dual to the complexity of a cutoff CFT. On the
CFT side we explicitly compute the differential entropy and holographic
complexity for the vacuum state and the excited state dual to conical AdS using
the kinematic space formalism. As a result, the boundary dual of the bulk first
law relates the first-order variations of differential entropy and complexity
to the variation of the scaling dimension of the excited state, which
corresponds to the matter Hamiltonian variation in the bulk. We also include
the variation of the central charge with associated chemical potential in the
boundary first law. Finally, we comment on the boundary dual of the first law
for the Wheeler-deWitt patch of AdS, and we propose an extension of our CFT
first law to higher dimensions.

High Energy Physics - Theory;General Relativity and Quantum Cosmology;Quantum Physics