How do I make page_action appear for specific pages?

For those looking for a way to handle subdomains, if you have a site with a subdomain such as blog.specificsite.com, or need to use wildcards, you can also use regex in this format

function checkForValidUrl(tabId, changeInfo, tab) 
{
    if(typeof tab != "undefined" && typeof tab != "null" )
    {
        // If the tabs URL contains "specificsite.com"...
        //This would work in the same way as *specificsite.com*, with 0 or more characters surrounding the URL.
        if (/specificsite[.]com/.test(tab.url)) 
        {
            // ... show the page action.
            chrome.pageAction.show(tabId);
        }
    }
};

// Listen for any changes to the URL of any tab.
chrome.tabs.onUpdated.addListener(checkForValidUrl);

to match the substring within the URL. It also helps with computation to do a null/undefined check to avoid additional exception handling.

http://code.google.com/chrome/extensions/pageAction.html
...says...

By default, a page action is hidden. When you show it, you specify the tab in which the icon should appear. The icon remains visible until the tab is closed or starts displaying a different URL (because the user clicks a link, for example).

So even if your tabid was valid it would dissapear pretty quick as your only running chrome.pageAction.show(tabId); once when the background page first gets run.
You need to check for changes to tabs in the background constantly because pageactions dont have matches/exclude_matches settings in the manifest like content scripts do (pity). So you have to check yourself and respond to changes.
If you want it to work for a specific site just change it to something like...

// Copyright (c) 2011 The Chromium Authors. All rights reserved.
// Use of this source code is governed by a BSD-style license that can be
// found in the LICENSE file.

// Called when the url of a tab changes.
function checkForValidUrl(tabId, changeInfo, tab) {
    // If the tabs url starts with "http://specificsite.com"...
    if (tab.url.indexOf('http://specificsite.com') == 0) {
        // ... show the page action.
        chrome.pageAction.show(tabId);
    }
};

// Listen for any changes to the URL of any tab.
chrome.tabs.onUpdated.addListener(checkForValidUrl);