  We begin an investigation of a new holographic covariant entropy bound (HCEB)
in gravity. This bound arises from Cauchy slice holography, a recently proposed
duality between the bulk gravity theory and a `boundary' theory that lives on
Cauchy slices. The HCEB is the logarithm of the maximum number of states of
this theory that can pass through a given cut $\sigma$ of a Cauchy slice
$\Sigma$ ($\sigma$ is thus a codimension-2 surface in the bulk). We show that
the bound depends only on the codimension-2 data on $\sigma$, and is thus
independent of the choice of slice $\Sigma$. For classical states, the HCEB
upper bounds the entanglement between two subregions of the boundary of
$\Sigma$.
  We calculate the bound explicitly in pure three-dimensional GR with negative
cosmological constant, where the Cauchy slice theory is the $T
\overline{T}$-deformation of the dual CFT. We find that the imaginary energy
eigenstates in the spectrum of the deformed theory play a crucial role for
obtaining a valid bound in Lorentzian signature. Our bound agrees with the area
of a surface at certain marginal and extremal surfaces, but differs elsewhere.
In particular, it exceeds the area by an arbitrarily large amount for
(anti)trapped surfaces, such as those that lie inside a black hole. Finally, we
discuss how these results can be used to write down tensor networks
corresponding to arbitrary Cauchy slices.
