Stopping interactive processes is a terrific way to break users' functionality expectations, because modern desktop applications do integrate through message buses tightly with the rest of the desktop environment. So, as tempting as your approach sounds, it's going to be a bad idea in practice. Not to mention that you really can't have an X window stop responding...
So, no, I don't think there's such a hack in any desktop environment I'm aware of.
Note that it's your browser that does something for you there - the right thing would be to approach what it's doing and make that stop specifically, if it's no use to you. I suspect it is - for example, a chat client in a browser tab might need to continuously receive and send status messages from a server, or your online word processing tool is not supposed to run out of sync, or your cache is being compressed or such.
Such backround jobs often are assigned low priorities by the software designers anyway (no idea whether that's the case here), and hence usually are only run when there's nothing better to do for your CPU. In other words, they would not significantly impact the performance. They might still consume power and thus reduce battery-supplied runtime, mostly because they might stop the operating system from putting a CPU core to sleep completely.
While that is still somewhat undesirable, you'd want to compare the actual impact to the negative effect that suspending/awakening the process might have (re-establishing all the TLS connections that got severed due to timeouts in TCP, large amounts of en-block data transfer at a time where high transmit power is needed by the WiFi card for high data rates, the simple fact that in the time that happens, you will have to wait, which means you're looking at a maximum brightness screen for longer - all effects that can eat into the energy savings!)
