Is cooked fish as perishable as raw fish?
Play it safe.
It has been said:

You probably will be ok, but the safer method is just to cook it the night before and refrigerate it. Toss the whole thing in an insulated lunch container, and you should have no problems. Even if the temperature creeps up into the danger zone, it shouldn't be there more than an hour or two before lunch time, and should still be cool enough not to cause an issue. –  JSM Aug 21 at 17:44

I happen to disagree with this advice, personally. While it may be OK, is this something worth trying to find out?
Here is an excerpt from an FDA research paper about pathogenic bacteria in food handling

Growth rates of pathogens are highly temperature dependent. Ordinarily, pathogenic bacteria growth is relatively slow at temperatures below 70°F (21.1°C). In most cases, growth is very slow below 50°F (10°C), and 40°F (4.4°C) is below the minimum growth temperature of most pathogenic bacteria, although there are some exceptions. On the other hand, pathogenic bacteria grow relatively fast at temperatures above 70°F (21.1°C).

Think of a real-world example, say a soda can. You get the can out of the machine at a temperature around 35°F-40°C, which is refrigeration temperature. You set it naked (i.e., no &quot;can cushy&quot;) on a table and just leave it there. Within 1-1.5 hours or so, it will be very close to room temperature, which on average is about 70°F to 73°F. That's when your fish will begin growing bacteria more rapidly, which it would continue to do until you are ready to eat it.
TL;DR
At the very least put and ice pack or other method of absorbing heat inside your lunchbox. Always play it safe when it comes to pathogens.