  We derive spectral sum rules in the shear channel for conformal field
theories at finite temperature in general $d\geq 3$ dimensions. The sum rules
result from the OPE of the stress tensor at high frequency as well as the
hydrodynamic behaviour of the theory at low frequencies. The sum rule states
that a weighted integral of the spectral density over frequencies is
proportional to the energy density of the theory. We show that the
proportionality constant can be written in terms the Hofman-Maldacena variables
$t_2, t_4$ which determine the three point function of the stress tensor. For
theories which admit a two derivative gravity dual this proportionality
constant is given by $\frac{d}{2(d+1)}$. We then use causality constraints and
obtain bounds on the sum rule which are valid in any conformal field theory.
Finally we demonstrate that the high frequency behaviour of the spectral
function in the vector and the tensor channel are also determined by the
Hofman-Maldacena variables.
