Most of Rishikesh's verified pet friendly hotels cluster around Tapovan, close to the river, and a genuinely high number of them confirm no pet charge at all.
Hotel DS Valley, Hotel Ishan, The Ekawa Ganga, and The Neeraj Ganga Heritage Palace all confirm pets stay free, with bowls provided at Hotel DS Valley and outdoor access at Hotel Ishan. Anand Kashi by the Ganges is the more specific option, welcoming small dogs under 10 kg and cats for ₹500 a night on the Ganga banks. Smol Door keeps pets in-room free of charge but excludes them from the pool, kitchen, and restaurant. The Jungle Mist Resorts and The Neeraj Ganga Cottages both sit closer to nature, bamboo cottages and a private Ganga Ghat respectively.
Given how many of these are free, the real differentiator is location and setting, riverside versus hillside, rather than cost.