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Why does Cases output an element only in the first of the following lines?

Cases[1, a -> 2 b, HoldPattern[a -> 2 b]]
(*a->2b*)
Cases[1, a -> 1/2 b, HoldPattern[a -> 1/2 b]] 
(**)
Cases[1, a -> π b, HoldPattern[a -> π b]] 
(**)
Cases[1, a -> c b, HoldPattern[a -> c b]] 
(**)

As an alternative to HoldPattern[Evaluate[...] you can use PatternSequence which evaluates its argument:

Cases[1, a -> 2 b, PatternSequence[a -> 2 b]],
 Cases[1, a -> 1/2 b, PatternSequence[a -> 1/2 b]],
 Cases[1, a -> π b, PatternSequence[a -> π b]],
 Cases[1, a -> c b, PatternSequence[a -> c b]]

a -> 2 b, a -> b/2, a -> b π, a -> b c

Alternatively, give the pattern a name:

Cases[1, a -> 2 b, p : (a -> 2 b)],
 Cases[1, a -> 1/2 b, p : (a -> 1/2 b)],
 Cases[1, a -> π b, p : (a -> π b)],
 Cases[1, a -> c b, p : (a -> c b)]

a -> 2 b, a -> b/2, a -> b π, a -> b c

Thanks to Kuba's comment now I see:
HoldPattern prevents the evaluation of its argument, so Cases don't match the element in the list a->b/2 that is

Times[Rational[1, 2], b]

with the unevaluated pattern

Times[b, Power[2, -1]]

Evaluate the argument of HoldPattern solves the problem:

Cases[1, a -> b/2, HoldPattern[Evaluate[a -> b/2]]]
(*a -> b/2*)

Thanks Kuba!

Here is another way by using Verbatim. Maybe it feels a bit less hacky.

 Cases[1, a -> 2 b, Verbatim[a -> 2 b]],
 Cases[1, a -> 1/2 b, Verbatim[a -> 1/2 b]],
 Cases[1, a -> π b, Verbatim[a -> π b]],
 Cases[1, a -> c b, Verbatim[a -> c b]]
 

a -> 2 b, a -> b/2, a -> b π, a -> b c