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* UK 10-year bond prices hit two-month low

* Rentokil Initial down on weaker demand outlook

* Hargreaves Lansdown falls on slower new client growth

* FTSE 100 down 1.1%, FTSE 250 off 0.6%

Oct 19 (Reuters) - The UK's FTSE 100 fell on Thursday as a rise in long-term government bond yields and uncertainty around the Middle East conflict kept investors risk-averse, while Rentokil Initial dropped on flagging weaker annual demand in North America.

The commodity-focused FTSE 100 was down 1.1% at 0838 GMT, while the mid-cap FTSE 250 slipped 0.6%.

British 10-year government bond prices fell to a new two-month low as they extended a sharp selloff driven by Wednesday's higher-than-expected inflation data and higher oil prices.

"Higher oil prices due to the geopolitical crisis in the Middle East will result in inflationary pressures, which is not helping the yield space," said Giles Coghlan, chief market analyst at GCFX Ltd.

Ten-year gilt yields rose as high as 4.726%, up more than 6 basis points on the day and the highest since Aug. 22.

Rentokil Initial tumbled 14% to the bottom of the FTSE 100 after the pest control firm forecast full-year performance slightly below the previous outlook due to softer demand in North America, its biggest market.

The broader industrial support services index fell 4.1%, touching its lowest level in over 11 months.

The heavyweight oil and gas index was down 1.0%, tracking lower oil prices, while industrial metal mining stocks lost 0.6% as copper prices slipped against a stronger dollar.

Precious metal miners added 0.5%, as demand rose for the safe-haven gold on growing tensions in the Middle East.

Hargreaves Lansdown reported a slowdown in new client growth for the first quarter. The investment platform's shares slipped 5.4%, while the investment banking and brokerage services index lost 0.9%.

Asset manager

Schroders

posted a fall in assets under management for the third quarter, taking its shares down 1.7%. (Reporting by Khushi Singh in Bengaluru; Editing by Sonia Cheema and Eileen Soreng)