No. You do not have to pay monthly for a gaming router. A gaming router is a one time purchase which do not need a monthly subscription.
But you need to pay for internet on a monthly basis as with the case of a normal router. Also, if your gaming router has a cyber security subscription, you may need to pay for that service on a yearly basis based on the service level agreement.
For example, the TP-Link AX6000 WiFi 6 Router(Archer AX6000) is a gaming router which also has support for the TPLINK Home care feature. This feature is a cyber security subscription based service which is provided by TPLINK.
So it can be understood that you need not pay monthly for a gaming router but only for the additional cyber security service which is provided by the router vendor.
Gaming router recommendation
The following are two gaming routers from two leading vendors – ASUS and TPLINK. More details on the respective routers can be found on Amazon.
1. The ASUS ROG Rapture WiFi 6 Gaming Router (GT-AX11000) is a Tri-Band 10 Gigabit Wireless Router. It has triple level game acceleration functionality With device, packet and game server prioritization.
More details on the router can be found on Amazon here – ASUS ROG Rapture WiFi 6 Gaming Router (GT-AX11000)
2. The TP-Link AC5400 Tri Band WiFi Gaming Router(Archer C5400X) supports 1.8GHz Quad-Core 64-bit CPU with game first Priority and Link aggregation.
It supports 8 Gigabit LAN ports and 1 Gigabit WAN port The Link Aggregation feature on the router can combine two ports to double connection speed. This is a Tri-band Wi-Fi router over one 2.4 GHz (1000Mbps) and two 5 GHz bands (2167Mbps), where one band is dedicated to gaming traffic.
More details on the router can be found on Amazon here – TP-Link AC5400 Tri Band WiFi Gaming Router(Archer C5400X)