Why did Peikoff say that modern scientists pit thermodynamics against hope?

I'm not positive, but I'm pretty sure he's referring to the notion of the "heat death" of the universe.

http://en.wikipedia.org/wiki/Heat_death_of_the_universe

A common mis-portrayal latched onto by some theists is that thermodynamics, specifically entropy is represented by deterioration. You build a house. If it is not actively maintained, it will crumble. Roads, automobiles, in fact just about any man-made material good breaks down and deteriorates over time.

Viewing that everything simply breaks down and deteriorates over the long haul is a pretty bleak forward outlook, which could easily be read as hopelessness an a general world-view.

This view, however, is derived not from the system as a whole, but looking at isolated sub-systems within it.

Just to clarify the science a little, the Wikipedia article cited by John ("Heat death of the universe") contains a section titled, "Current status," which nicely summarizes the scientific issues regarding the proper scientific application of the Second Law of Thermodynamics (pertaining to entropy). As Dream Weaver seems to be noting, one cannot properly apply the Second Law only to the Earth, because the Earth continues to receive huge quantities of energy from the Sun. One must include the Sun in the analysis, as well (and possibly effects from the rest of the Milky Way and beyond, also). In response to the Sun's energy, it is entirely possible for the Earth to continue to evolve toward increasing "order" (decreasing entropy) rather than decreasing order, for as long as the Sun continues to burn. The Sun, meanwhile, is widely predicted to burn out long before the universe as a whole ever comes close to reaching thermodynamic equilibrium and a correspondingly maximum entropy throughout (if it ever does). Those who worry about a predicted thermodynamic end of the universe (which is not necessarily endorsed by science, as the Wikipedia article explains) ought to be more concerned about the expected finite lifespan of the Sun, without which Earthly life as we know it would be unable to survive. Fortunately, we still have many billions years to go before the end of the Sun kills us all.