  Hydrodynamical interaction in circumbinary discs (CBDs) plays a crucial role
in various astrophysical systems, ranging from young stellar binaries to
supermassive black hole binaries in galactic centers. Most previous simulations
of binary-disc systems have adopted locally isothermal equation of state. In
this study, we use the grid-based code $\texttt{Athena++}$ to conduct a suite
of two-dimensional viscous hydrodynamical simulations of circumbinary accretion
on a cartesian grid, resolving the central cavity of the binary. The gas
thermodynamics is treated by thermal relaxation towards an equilibrium
temperature (based on the constant$-\beta$ cooling ansatz, where $\beta$ is the
cooling time in units of the local Keplerian time). Focusing on equal mass,
circular binaries in CBDs with (equilibrium) disc aspect ratio $H/R=0.1$, we
find that the cooling of the disc gas significantly influences the binary
orbital evolution, accretion variability, and CBD morphology, and the effect
depends sensitively on the disc viscosity prescriptions. When adopting a
constant kinematic viscosity, a finite cooling time ($\beta \gtrsim 0.1$) leads
to binary inspiral as opposed to outspiral and the CBD cavity becomes more
symmetric. When adopting a dynamically varying $\alpha-$viscosity, binary
inspiral only occurs within a narrow range of cooling time (corresponding to
$\beta$ around 0.5).
