Change Cart total price in WooCommerce

This does not answer this question. Loic's does. This is another way of doing it to show a line item of 10% off:

function prefix_add_discount_line( $cart ) {

  $discount = $cart->subtotal * 0.1;

  $cart->add_fee( __( 'Down Payment', 'yourtext-domain' ) , -$discount );

}
add_action( 'woocommerce_cart_calculate_fees', 'prefix_add_discount_line' );

Since Woocommerce 3.2+ it does not work anymore with the new Class WC_Cart_Totals ...

New answer: Change Cart total using Hooks in Woocommerce 3.2+

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First woocommerce_cart_total hook is a filter hook, not an action hook. Also as wc_price argument in woocommerce_cart_total is the formatted price, you will not be able to increase it by 10%. That's why it returns zero.

Before Woocommerce v3.2 it works as some WC_Cart properties can be accessed directly

You should better use a custom function hooked in woocommerce_calculate_totals action hook
this way:

// Tested and works for WooCommerce versions 2.6.x, 3.0.x and 3.1.x
add_action( 'woocommerce_calculate_totals', 'action_cart_calculate_totals', 10, 1 );
function action_cart_calculate_totals( $cart_object ) {

    if ( is_admin() && ! defined( 'DOING_AJAX' ) )
        return;

    if ( !WC()->cart->is_empty() ):
        ## Displayed subtotal (+10%)
        // $cart_object->subtotal *= 1.1;

        ## Displayed TOTAL (+10%)
        // $cart_object->total *= 1.1;

        ## Displayed TOTAL CART CONTENT (+10%)
        $cart_object->cart_contents_total *= 1.1;

    endif;
}

Code goes in function.php file of your active child theme (or theme) or also in any plugin file.

Is also possible to use WC_cart add_fee() method in this hook, or use it separately like in Cristina answer.