  Planets can excite density waves and open annular gas gaps in protoplanetary
disks. The depth of gaps is influenced by the evolving angular momentum carried
by density waves. While the impact of radiative cooling on the evolution of
density waves has been studied, a quantitative correlation to connect gap depth
with the cooling timescale is lacking. To address this gap in knowledge, we
employ the grid-based code Athena++ to simulate disk-planet interactions,
treating cooling as a thermal relaxation process. We establish quantitative
dependences of steady-state gap depth (Eq. 36) and width (Eq. 41) on planetary
mass, Shakura-Sunyaev viscosity, disk scale height, and thermal relaxation
timescale $(\beta)$. We confirm previous results that gap opening is the
weakest when thermal relaxation timescale is comparable to local dynamical
timescale. Significant variations in gap depth, up to an order of magnitude,
are found with different $\beta$. In terms of width, a gap is at its narrowest
around $\beta=1$, approximately $10\%$ to $20\%$ narrower compared to the
isothermal case. When $\beta\sim100$, it can be $\sim20\%$ wider, and higher
viscosity enhances this effect. We derive possible masses of the gas
gap-opening planets in AS 209, HD 163296, MWC 480, and HL Tau, accounting for
the uncertainties in local thermal relaxation timescale.
