use strict;
use warnings;
my ($outer_hash, $new_hash);

$outer_hash->{backupsize} = [{
                              'totalsize' => '89',
                              'hostname' => 'aaaa',
                              'application' => 'IDB'
                            },
                            {
                              'application' => 'IDB',
                              'hostname' => 'aaaa',
                              'totalsize' => '32770'
                            },
                            {
                              'application' => 'SAP',
                              'hostname' => 'bbbb',
                              'totalsize' => '14'
                            }];
for my $entry(@{ $outer_hash->{backupsize} }){
    next unless $entry->{hostname} and $entry->{application} and $entr
+y->{totalsize};
    $new_hash->{$entry->{hostname}} {$entry->{application}} += $entry-
+>{totalsize};
}

for my $h (sort keys %$new_hash){
    print "HOST:$h:\n";
    for my $app(keys %{ $new_hash->{$h}}){
        print "\t",$new_hash->{$h}{$app},"\t$app\n";
    }
}
[download]

OUTPUT:

HOST:aaaa:
    32859    IDB
HOST:bbbb:
    14    SAP
[download]