The properties of molecular gas, the fuel that forms stars, inside the cavity of the circumnuclear disk (CND) are not well constrained. We present results of a velocity-resolved submillimeter scan (~480–1250 GHz) and [C II] 158 μm line observations carried out with Herschel/HIFI toward Sgr A^*; these results are complemented by a ~2′ × 2′ ¹²CO (J = 3−2) map taken with the IRAM 30 m telescope at ~7″ resolution. We report the presence of high positive-velocity emission (up to about +300 km s⁻¹) detected in the wings of ¹²CO J = 5−4 to 10−9 lines. This wing component is also seen in H₂O (1_1,0−1_0,1), a tracer of hot molecular gas; in [C II]158 μm, an unambiguous tracer of UV radiation; but not in [C I] 492, 806 GHz. This first measurement of the high-velocity ¹²CO rotational ladder toward Sgr A^* adds more evidence that hot molecular gas exists inside the cavity of the CND, relatively close to the supermassive black hole (<1 pc). Observed by ALMA, this velocity range appears as a collection of ¹²CO (J = 3−2) cloudlets lying in a very harsh environment that is pervaded by intense UV radiation fields, shocks, and affected by strong gravitational shears. We constrain the physical conditions of the high positive-velocity CO gas component by comparing with non-LTE excitation and radiative transfer models. We infer T_k ≃ 400 K–2000 K for n_H ≃ (0.2−1.0) × 10⁵ cm⁻³. These results point toward the important role of stellar UV radiation, but we show that radiative heating alone cannot explain the excitation of this ~10−60 M_⊙ component of hot molecular gas inside the central cavity. Instead, strongly irradiated shocks are promising candidates.