I can think of several possible issues here.

(1) Firefox may be losing track of your network connection for some reason and going offline. There is a setting in Firefox to prevent it from doing that.

(i) In a new tab, type or paste about:config in the address bar and press Enter. Click the button promising to be careful.

(ii) In the filter box, type or paste offline and pause while the list is filtered

(iii) Look for network.manage-offline-status and make sure it is set to False (double-click to switch the value between true and false).

Does that make any difference?

(2) Your Mac may be saving power by putting your network connection to sleep.

(3) Your router may be flaking out for some reason.

(4) Your ISP may be going offline for its own reasons. Maybe you can test this theory by very quietly playing an internet radio station in the background to ensure that your connection is never idle.

(1) It is too complicated for me (2) I never changed that setting; the problem started with the new router (3) ?? (4) Correct: with radio on, never iddle. Thank you for your help

I know it's alien territory the first time... If you try to work through the steps on #1, where do you run into problems?

network.manage-offline-status is already set to False. Any way to troubleshoot items 3 and 4 ?

If it's a wireless router and you do not live miles out in the woods, you may be getting some connections that are competing with yours, or interference from your neighbors.

There are some steps you can take to make your router less visible and less subject to attempted connections. For example, stop broadcasting your SSID, turn off plug-and-play configuration, and use one of the newer methods of encryption, such as WPA2.

For possible interference, if you look at the available networks using your Mac or a smartphone, make sure you are using a different channel than the nearest strong signals. Channels 1, 6, and 11 do not interfere with one another; they are far enough apart to avoid interference. However, if those are taken, you could try splitting the difference between two weaker ones.

In general, these topics are beyond the scope of this forum. If you search around, you can find some good forums on networking issues.