Nothing to do with grep in particular, or constants.

use constant REGEXP => qr(a);
print grep { REGEXP } qw(a b c);
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... is equivalent to:

my @tmp;
foreach (qw(a b c)) {
   push @tmp, $_ if qr(a);
}
print @tmp;
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See the problem now? if qr(a) is always true. if qr(anything) is always true.

Try:

print grep { $_ =~ REGEXP } @list;
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