  We develop a theory of heat transfer induced by thermal charge fluctuations
in two-dimensional electron double layers. We consider pristine systems
comprised of identical layers, and focus on the regime of sufficiently high
temperatures and interlayer distances $d$, where the relevant charge
fluctuations may be described using the hydrodynamic approach. In this limit
heat transfer is dominated by the plasmon resonances. For systems with
Galilean-invariant electron dispersion the interlayer thermal conductance
$\varkappa$ is proportional to the kinematic viscosity of the electron liquid,
and decreases as $1/d^4$. In the absence of Galilean invariance $\varkappa
\propto \sigma/d^3$, where $\sigma$ is the intrinsic conductivity of the
liquid. This strong enhancement can be traced to a drastically different
broadening of plasmon resonances in systems with and without Galilean
invariance.
