@a5n
Typically +/- 10% but you would not want this for a direct to battery connection as at the lower end you may end up pulling more current than you would wish for (and your typical 12v lead-acid battery has lots of that, primed and ready to weld or fry) and the ripple at the higher end (eg for charging) is not friendly either.
You can purchase 12v-12v dc regulators that would aim for (at most) a 5% tolerance on 12vdc output but they are usually much better than that. This device would be placed between the battery output and the router. They are often marketed at the automotive sector, for obvious reasons.
The input tolerance on these tends to be quite wide (say 10v to 36v) to produce a nice and stable 12vdc output, even when the source battery is being charged. They do consume some power themselves and typically come potted into an external heatsink.
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So in short, the only thing you should ever connect to the router is a regulated dc source, never direct to a battery.👍
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