  We examine the abundance of molecular hydrogen (H2) in the spectra of gamma
ray burst afterglows (GRBs). In nearby galaxies H2 traces the cold neutral
medium (CNM) and dense molecular star-forming interstellar gas. Though H2 is
detected in at least half of all sightlines towards hot stars in the Magellanic
Clouds and in ~25% of damped Lya systems toward quasars, it is not detected in
any of the five GRB environments with a similar range of neutral hydrogen
column and metallicity. We detect no vibrationally-excited H2 that would imply
the GRB itself has photodissociated its parent molecular cloud, so such models
are ruled out unless the parent cloud was <~4 pc in radius and was fully
dissociated prior to the spectroscopic observations, or the star escaped its
parent cloud during its main-sequence lifetime. The low molecular fractions for
the GRBs are mysterious in light their large column densities of neutral H and
expectations based on local analogs, i.e. 30 Doradus in the LMC. This
surprising lack of H2 in GRB-DLAs indicates that the destruction processes that
suppress molecule formation in the LMC and SMC are more effective in the GRB
hosts, most probably a combination of low metallicity and an FUV radiation
field 10--100 times the Galactic mean field. These inferred conditions place
strong constraints on the star forming regions in these early galaxies.
