CHARLOTTE, N.C. — Time Warner Cable will soon charge its 15 million customers for watching sports as the major cable provider tries compensating for the rising costs of broadcasting local channels and athletic programming. Effective Jan. 1, the cable company will charge subscribers a new $2.75-per-month fee for sports programming. Other changes include a 9-month-old broadcasting TV surcharge increasing from $2.25 to $2.76 per month, and an $8-per-month Internet modem lease, up from $5.99. Subscribers who pay for HBO movie channels selectively on an “a la carte” basis will see those rates increase on their bills from $14.99 to $16.99.