following the answers given by Pantev, I will give you more example of canonical topology on some category(universally strict epimorphism)
if C is an abelian category or a topos, then canonical topology consists of all epimorphism
if C is a quasi abelian category,then canonical topology consists of all strict epimorphism(Note that strict epimorphism is subcanonical in general)
if C is a category of associative unital k-algebras(opposite category of affine schemes,not necessarily commutative). Canonical topology consists of all strict epimorphism which are precisely surjective morphism of algebras. In this case, univerally strict epimorphism coincides with strict epimorphism.
Rosenberg has a very detailed treatment for a category and 2-category(taken as category of spaces)with canonical topology(he called right exact structure).It is in MPIM preprint series," Homological algebra of noncommutative 'space' I"
What pantev mentioned is related to my answers in another question:
Does sheafification preserve sheaves for a different topology?
The effective descent topology is finer that fpqc topology, fppf topology,smooth topology(in Kontsevich-Rosenberg sense)
The descent topology on category of affine schemes(not necessarily commutative) coincides with subcanonical topology
The reference is Orlov's paper and Kontsevich-Rosenberg MPIM preprint series. Noncommutative stack and Noncommutative grassmannian and related construction