Why should the temperature be maintained at 0–5 °C in a diazotisation?
The temperature at which many reactions are carried out is often arbitrary, with certain values being favoured due to the ease at which cooling baths can be prepared. As an example  -78°C is highly prevalent, and a question about this 'magic temperature' has been asked here previously.

In the case of reactions forming diazonium salts, there has been a common rule of thumb that they must be kept somewhere between 0 - 5°C, which is even quoted on Wikipedia:


  In aqueous solution diazonium salts are unstable at temperatures above 5 °C  (Ref: Wikipedia)


At temperatures above 5 °C, diazonium salts (in aqueous solution) have been known to decompose, often explosively, to liberate nitrogen gas, temperatures too low however can cause crystallisation, which causes issues later on (diazonium salts should never be dried upon isolation - they explode). 

Although people often keep to this rule of thumb, theres no real body of evidence for its generality (i.e. different diazonium species will have vastly different decomposition temperatures). 

Often the only time people carry out in-depth studies of a particular reaction is when the reaction needs to be scaled up, such as in the the pharmaceutical industry. Process chemists go to extraordinary lengths to study their reactions to avoid any possible safety issues (or even unintended side reactions).  In Organic Process Research &amp; Development 2004, 8, 1059−1064, a team of process chemists at Lundbeck carried out extensive thermal studies on a diasotisation reaction they were doing as part of a Sandmeyer reaction, and they found no issues well above 5 °C (though as a precaution they still kept the reaction below 15 °C).