To investigate the presence of small scale structure in the spatial distribution of H2 molecules we have undertaken repeated FUSE UV observations of the runaway O9.5V star, HD34078. In this paper we present five spectra obtained between January 2000 and October 2002. These observations reveal an unexpectedly large amount of highly excited H2. Column densities for H2 levels from (v = 0, J = 0) up to (v = 0, J = 11) and for several v = 1 and v = 2 levels are determined. These results are interpreted in the frame of a model involving essentially two components: i) a foreground cloud (unaffected by HD34078) responsible for the H2 (J = 0, 1), CI, CH, CH+ and CO absorptions; ii) a dense layer of gas (n = 10E4 cm-3) close to the O star and strongly illuminated by its UV flux which accounts for the presence of highly excited H2. Our model successfully reproduces the H2 excitation, the CI fine-structure level populations as well as the CH, CH+ and CO column densities. We also examine the time variability of H2 absorption lines tracing each of these two components. From the stability of the J = 0, 1 and 2 damped H2 profiles we infer a 3 sigma upper limit on column density variations Delta(N(H2))/N(H2) of 5% over scales ranging from 5 to 50 AU. This result clearly rules out any pronounced ubiquitous small scale density structure of the kind apparently seen in HI. The lines from highly excited gas are also quite stable (equivalent to Delta(N)/N <= 30%) indicating i) that the ambient gas through which HD34078 is moving is relatively uniform and ii) that the gas flow along the shocked layer is not subject to marked instabilities
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