Instead of promoting the the definitions of the constants to compile time, you can instead load the module at run time:

use Config::Std;
use Readonly;
Readonly my $DEFAULT_LOG_LEVEL   => q{warn};
Readonly my $DEFAULT_LOG_FILE    => $PROGRAM_NAME . q{.log};
Readonly my $DEFAULT_CONFIG_FILE => $PROGRAM_NAME . q{.cfg};
read_config $DEFAULT_CONFIG_FILE => my %config;

require Log::StdLog;
Log::StdLog->import({
    level => ($config{'log'}{'level'} || $DEFAULT_LOG_LEVEL), 
    file  => ($config{'log'}{'file' } || $DEFAULT_LOG_FILE )
});
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Note that this also means that import of various subs and symbols happens at run time, so it might or might not work for you.