So, the reason that the split approach fails is that it is very sensitive to changes in input structure. If we examine the corrected input stream, you'll note that it is now valid JSON, so the script

use strict;
use warnings;
use JSON;
use Data::Dumper;

$_ = '{"temp":75.50,"tmode":2,"fmode":0,"override":0,"hold":0,"t_cool"
+:75.00,"tstate":0,"fstate":0,"time":{"day":4,"hour":13,"minute":49},"
+t_type_post":0}';

print Dumper decode_json ($_);
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outputs

$VAR1 = {
          'fstate' => 0,
          't_cool' => '75',
          'time' => {
                      'hour' => 13,
                      'minute' => 49,
                      'day' => 4
                    },
          'fmode' => 0,
          'tmode' => 2,
          'temp' => '75.5',
          'hold' => 0,
          'override' => 0,
          'tstate' => 0,
          't_type_post' => 0
        };
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Rather than having to fight writing a parser, the module does the heavy lifting, and your script could be rewritten perhaps like:

use strict;
use warnings;
use JSON;

open (my $fh, '<', '/home/julian/tstatcollect') or die "Open failed:$!
+";
while (<$fh>) {

    chomp;
    my $obj = decode_json($_);
    
    print "Temperature: $obj->{temp}\n";
    print "Tmode:       $obj->{tmode}\n";
    print "Fmode:       $obj->{fmode}\n";
    print "Override:    $obj->{override}\n";
    print "Hold:        $obj->{hold}\n";
    print "tcool:       $obj->{t_cool}\n";
    print "tstate:      $obj->{tstate}\n";
    print "fstate:      $obj->{fstate}\n";
    print "day:         $obj->{time}{day}\n";
    print "hour:        $obj->{time}{hour}\n";
    print "minute:      $obj->{time}{minute}\n";
    print "t_type_post  $obj->{t_type_post}\n";
    
    print "________\n";
}
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which outputs

Temperature: 75.5
Tmode:       2
Fmode:       0
Override:    0
Hold:        0
tcool:       75
tstate:      0
fstate:      0
day:         4
hour:        13
minute:      49
t_type_post  0
________
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