The Galactic center is the closest region in which we can study star formation under extreme physical conditions like those in high-redshift galaxies. We measure the temperature of the dense gas in the central molecular zone (CMZ) and examine what drives it. We mapped the inner 300 pc of the CMZ in the temperature-sensitive J = 3-2 para-formaldehyde (p-H$_2$CO) transitions. We used the $3_{2,1} - 2_{2,0} / 3_{0,3} - 2_{0,2}$ line ratio to determine the gas temperature in $n sim 10^4 - 10^5 $cm$^{-3}$ gas. We have produced temperature maps and cubes with 30 and 1 km/s resolution and published all data in FITS form. Dense gas temperatures in the Galactic center range from ~60 K to > 100 K in selected regions. The highest gas temperatures T_G > 100 K are observed around the Sgr B2 cores, in the extended Sgr B2 cloud, the 20 km/s and 50 km/s clouds, and in The Brick (G0.253+0.016). We infer an upper limit on the cosmic ray ionization rate ${zeta}_{CR} < 10^{-14}$ 1/s. The dense molecular gas temperature of the region around our Galactic center is similar to values found in the central regions of other galaxies, in particular starburst systems. The gas temperature is uniformly higher than the dust temperature, confirming that dust is a coolant in the dense gas. Turbulent heating can readily explain the observed temperatures given the observed line widths. Cosmic rays cannot explain the observed variation in gas temperatures, so CMZ dense gas temperatures are not dominated by cosmic ray heating. The gas temperatures previously observed to be high in the inner ~75 pc are confirmed to be high in the entire CMZ.
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