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Hydrodynamical interactions between binaries and circumbinary disks (CBDs) play an important role in a variety of astrophysical systems, from young stellar binaries to supermassive black hole binaries. Previous simulations of CBDs have mostly employed locally isothermal equation of state. We carry out two-dimensional viscous hydrodynamic simulations of CBDs around equal-mass, circular binaries, treating the gas thermodynamics by thermal relaxation towards equilibrium temperature (the constant-$β$ cooling ansatz, where $β$ is the cooling time in units of the local Keplerian time). As an initial study, we use the grid-based code Athena++ on a polar grid, covering an extended disk outside the binary co-orbital region. We find that with a longer cooling time, the accretion variability is gradually suppressed, and the morphology of the CBD becomes more symmetric. The disk also shows evidence of hysterisis behavior depending on the initial conditions. Gas cooling also affects the rate of angular momentum transfer between the binary and the CBD, where given our adopted disk thickness and viscosity ($H/r\sim 0.1$ and $α\sim 0.1$), the binary orbit expands while undergoing accretion for most $β$ values between 0 and 4.0 except over a narrow range of intermediate $β$ values. The validity of using polar grid excising the central domain is also discussed.