We present ISO observations of several \Hd pure-rotational lines (from S(0) to S(5)) towards a sample of 16 molecular clouds distributed along the central 500 pc of the Galaxy. We also present \CuoO and \utCO \juce and \jdu observations of these sources made with the IRAM-30m telescope. With the CO data we derive \Hd densities of 103.5-4.0 \cmmt and \Hd column densities of a few 1022 \cmmd. We have corrected the \Hd data for 30 magnitudes of visual extinction using a self-consistent method. In every source, we find that the \Hd emission exhibits a large temperature gradient. The S(0) and S(1) lines trace temperatures (T) of 150 K while the S(4) and S(5) lines indicate temperatures of 600 K. The warm \Hd column density is typically 1-2* 1022 \cmmd, and is predominantly gas with T=150 K. This is the first direct estimate of the total column density of the warm molecular gas in the Galactic center region. These warm \Hd column densities represent a fraction of 30 % of the gas traced by the CO isotopes emission. The cooling by \Hd in the warm component is comparable to that by CO. Comparing our \Hd and CO data with available ammonia (\NHt) observations from literature one obtains relatively high \NHt abundances of a few 10-7 in both the warm and the cold gas. A single shock or Photo-Dissociation Region (PDR) cannot explain all the observed \Hd lines. Alternatives for the heating mechanisms are discussed.