You could try the following to see if it is the modem. Power down the computer, disconnect the modem. Start the computer. Then press, Ctrl-Shift-Esc to bring up the Task Manager. Just leave that run for about an hour. You can switch between the Processes and Performance tabs. Note the fan's cutting in and out.
Then, power down. Connect the modem. If you want to wait a while, do so. Power up. Again press Ctrl-Shift-Esc. For an hour note the fan cut in and what processes are running as well CPU usage when you go to the performance tab. If CPU usage is higher, it will normally run hotter.
If there is no change, then go online keeping the task manager operating in the background and watch the CPU usage - both when you are using your browser or when you have logged off.