The sudden and repeated extinction events of trilobites provides clues about the climate changes in the Cambrian.
All of the continents were close to the equator and the trilobites were adapted to warm waters presumably. It has been suggested that the extinction of the trilobites was associated with a cooling of the ocean waters
This hypothesis is supported by the fact that it was only the deeper dwelling trilobites that survived the extinctions. This is probably because they were already adapted to cold conditions since they lived in deep (cold) waters. Nonetheless, the case of temperature change as the cause remains unproven.
What is particularly problematic about any cooling idea to explain the exitinction is evidence that suggest that atmosphereic CO2 was much higher in the early Paleozoic Era. This evidence is in the form of various mineral types whose presence is a sensitive indicator of atmospheric CO2 levels.
Aragonite is a form of calcite (like calcium carbonate). It is very susceptable to dissolution (it will dissolve readily) if atmospheric pCO2 is high. When the line falls below the Aragonite threshold (hatched area) it implies lower pCO2 levels in the atmosphere.
If you look at the diagram above you can see that in the early part of the Paleozoic (CAM. ORD., SIL.) there is virtually no aragonite only Calcite.
This diagram suggests that although that atmospheric CO2 was lower in the early part of the Cambrian it may have been increasing by the end of the Cambrian (see diagram above) .
Perhaps there were short periods of decreased pCO2 that coincided with the cooling and extinctions. This evidence does not yet exist but is a topic of considerable interest among Earth scientists.